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1</head><body class="min-h-screen flex flex-col font-sans"><a href="#main-content" class="skip-to-main">Skip to main content</a><header class="sticky top-0 z-50 w-full border-b bg-background/95 backdrop-blur supports-[backdrop-filter]:bg-background/60"><nav class="mx-auto flex max-w-7xl items-center justify-between p-3 sm:p-6 lg:px-8" aria-label="Global navigation"><div class="flex lg:flex-1"><a class="-m-1.5 p-1.5 font-bold text-lg sm:text-xl focus-visible-ring flex items-center gap-2" href="/"><span class="sr-only">PerfectFinder</span><svg viewBox="0 0 100 100" fill="none" xmlns="http://www.w3.org/2000/svg" class="w-6 h-6 sm:w-8 sm:h-8"><rect x="0" y="0" width="100" height="100" rx="22" fill="#6366F1"></rect><path d="M 25 50 C 25 38, 32 30, 42 38 C 48 43, 52 50, 50 50 C 48 50, 52 57, 58 62 C 68 70, 75 62, 75 50 C 75 38, 68 30, 58 38 C 52 43, 48 50, 50 50 C 52 50, 48 57, 42 62 C 32 70, 25 62, 25 50 Z" stroke="white" stroke-width="7" stroke-linecap="round" stroke-linejoin="round" fill="none"></path></svg><span class="bg-gradient-to-r from-indigo-500 via-purple-500 to-purple-400 bg-clip-text text-transparent">PerfectFinder.ai</span></a></div><div class="flex lg:hidden"><button type="button" class="-m-2.5 inline-flex items-center justify-center rounded-md p-2.5 text-foreground focus-visible-ring" aria-expanded="false" aria-label="Toggle navigation menu"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-menu h-6 w-6" aria-hidden="true"><line x1="4" x2="20" y1="12" y2="12"></line><line x1="4" x2="20" y1="6" y2="6"></line><line x1="4" x2="20" y1="18" y2="18"></line></svg></button></div><div class="hidden lg:flex lg:gap-x-8"><a class="text-sm font-semibold leading-6 text-foreground hover:text-primary focus-visible-ring rounded px-2 py-1" href="/how-it-works">How It Works</a><div class="relative group"><button class="text-sm font-semibold leading-6 text-foreground hover:text-primary focus-visible-ring rounded px-2 py-1">Solutions</button><div class="absolute left-0 mt-2 w-56 rounded-2xl bg-popover shadow-lg opacity-0 invisible group-hover:opacity-100 group-hover:visible transition-all duration-200"><div class="p-2"><a class="block px-4 py-2 text-sm text-foreground hover:bg-accent rounded-xl focus-visible-ring" href="/for-listing-platforms">For Listing Platforms</a><a class="block px-4 py-2 text-sm text-foreground hover:bg-accent rounded-xl focus-visible-ring" href="/for-consumers">For Consumers</a><a class="block px-4 py-2 text-sm text-foreground hover:bg-accent rounded-xl focus-visible-ring" href="/for-marketplaces">For Marketplaces</a><a class="block px-4 py-2 text-sm text-foreground hover:bg-accent rounded-xl focus-visible-ring" href="/for-compliance">For Compliance</a></div></div></div><a class="text-sm font-semibold leading-6 text-foreground hover:text-primary focus-visible-ring rounded px-2 py-1" href="/demos">Demos</a><a class="text-sm font-semibold leading-6 text-foreground hover:text-primary focus-visible-ring rounded px-2 py-1" href="/roi">ROI</a><a class="text-sm font-semibold leading-6 text-foreground hover:text-primary focus-visible-ring rounded px-2 py-1" href="/patent">The Patent</a><a class="text-sm font-semibold leading-6 text-foreground hover:text-primary focus-visible-ring rounded px-2 py-1" href="/about">About</a></div><div class="hidden lg:flex lg:flex-1 lg:justify-end lg:gap-4 lg:items-center"><a href="/contact"><button class="inline-flex items-center justify-center whitespace-nowrap font-medium ring-offset-background transition-all focus-visible:outline-none focus-visible:ring-2 focus-visible:ring-ring focus-visible:ring-offset-2 disabled:pointer-events-none disabled:opacity-50 bg-gradient-to-r from-indigo-500 via-purple-500 to-purple-400 text-white hover:shadow-xl hover:shadow-purple-500/50 hover:scale-105 shadow-lg h-14 roun
1ded-2xl px-8 text-base">Partner with us</button></a></div></nav></header><main id="main-content" class="flex-1"><div class="flex flex-col"><section class="px-6 pt-14 lg:px-8 pb-16 sm:pb-24"><div class="mx-auto max-w-4xl text-center"><h1 class="text-4xl font-bold tracking-tight text-foreground sm:text-5xl">How It Works</h1><p class="mt-6 text-lg leading-8 text-muted-foreground max-w-2xl mx-auto">From your natural-language needs to a provably optimal result in milliseconds</p></div></section><section class="py-16 sm:py-24 bg-muted/30"><div class="mx-auto max-w-7xl px-6 lg:px-8"><div class="mx-auto max-w-2xl text-center mb-16"><h2 class="text-3xl font-bold tracking-tight sm:text-4xl mb-4">Four steps to optimal</h2><p class="text-lg text-muted-foreground">The system transforms your intent into a dominance-aware decision</p></div><div class="mx-auto max-w-5xl space-y-12"><div class="flex flex-col md:flex-row gap-6 items-start"><div class="flex-shrink-0 w-16 h-16 rounded-2xl bg-primary text-primary-foreground flex items-center justify-center text-2xl font-bold">1</div><div class="flex-1"><div class="flex items-start gap-3 mb-3"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-message-square h-6 w-6 text-primary flex-shrink-0 mt-1"><path d="M21 15a2 2 0 0 1-2 2H7l-4 4V5a2 2 0 0 1 2-2h14a2 2 0 0 1 2 2z"></path></svg><h3 class="text-2xl font-bold">Parse natural language â constraints + criteria</h3></div><p class="text-muted-foreground mb-4">You describe what you need in plain English: "Under $1,000, for software development, must have at least 16GB RAM, 512GB SSD, under 1.5kg, 2 USB-C ports, Wi-Fi 6E, deliver by Nov 1. I prefer gray or blackâgray is better than black. I prefer Intel to AMD processor, and I wouldn't like a refurbished laptop."</p><div class="grid md:grid-cols-2 gap-6"><div class="rounded-2xl border bg-card text-card-foreground shadow-lg"><div class="flex flex-col space-y-1.5 p-6"><h3 class="font-semibold tracking-tight text-lg">Mandatory Constraints</h3><p class="text-sm text-muted-foreground">All candidates must satisfy these</p></div><div class="p-6 pt-0"><div class="space-y-2"><div class="flex items-start gap-2 text-sm"><span class="text-green-600 dark:text-green-400 mt-0.5">â</span><span>max price: $1000</span></div><div class="flex items-start gap-2 text-sm"><span class="text-green-600 dark:text-green-400 mt-0.5">â</span><span>min ram: 16</span></div><div class="flex items-start gap-2 text-sm"><span class="text-green-600 dark:text-green-400 mt-0.5">â</span><span>min storage: 512</span></div><div class="flex items-start gap-2 text-sm"><span class="text-green-600 dark:text-green-400 mt-0.5">â</span><span>max weight: 1.5</span></div><div class="flex items-start gap-2 text-sm"><span class="text-green-600 dark:text-green-400 mt-0.5">â</span><span>usb c ports: 2</span></div><div class="flex items-start gap-2 text-sm"><span class="text-green-600 dark:text-green-400 mt-0.5">â</span><span>wifi: 6E</span></div><div class="flex items-start gap-2 text-sm"><span class="text-green-600 dark:text-green-400 mt-0.5">â</span><span>delivery date: 2025-11-01</span></div></div></div></div><div class="rounded-2xl border bg-card text-card-foreground shadow-lg"><div class="flex flex-col space-y-1.5 p-6"><h3 class="font-semibold tracking-tight text-lg">Preference Hierarchy</h3><p class="text-sm text-muted-foreground">Your explicit preferences organized by priority</p></div><div class="p-6 pt-0"><div class="space-y-3"><div class="border-2 border-primary/30 bg-primary/5 rounded-lg p-3"><div class="text-xs font-semibold text-primary mb-2">PREFERENCE LAYER (PARTIAL ORDER)</div><div class="space-y-2 text-sm"><div class="flex items-center gap-2"><div class="w-2 h-2 rounded-full bg-primary"></div><span>Gray > Black</span></div><div class="flex items-center gap-2"><div class="w-2 h-2 rounded-full bg-primary"></div><span>Intel > AMD</span></div><div class="flex items-center gap-2"><div class="w-2 h-2 rounded-full bg-primary"></div><span>New > Refurbished</span></div></div></div></div><p class="text-xs text-muted-foreground mt-3 italic">These preferences form a partial order (DAG) - they are independent comparisons, not strict levels</p></div></div><div class="md:col-span-2"><div class="rounded-2xl text-card-foreground shadow-lg border-dashed border-2 border-purple-300 bg-purple-50/30 dark:bg-purple-950/20"><div class="flex flex-col space-y-1.5 p-6"><h3 class="font-semibold tracking-tight text-lg text-purple-700 dark:text-purple-300">Implicit Domain Preferences</h3><p class="text-sm text-muted-foreground">Universal domain preferences used to break ties between equally optimal results (lowest priority)</p></div><div class="p-6 pt-0"><div class="grid md:grid-cols-2 gap-4"><div class="flex items-center gap-2 text-sm"><div class="w-2 h-2 rounded-full bg-purple-500"></div><span class="font-medium">Higher Rating</span><span class="text-xs text-muted-foreground">â Tie-breaker preference</span></div><div class="flex items-center gap-2 text-sm"><div class="w-2 h-2 rounded-full bg-purple-500"></div><span class="font-medium">More RAM</span><span class="text-xs text-muted-foreground">â Tie-breaker preference</span></div><div class="flex items-center gap-2 text-sm"><div class="w-2 h-2 rounded-full bg-purple-500"></div><span class="font-medium">More Storage</span><span class="text-xs text-muted-foreground">â Tie-breaker preference</span></div><div class="flex items-center gap-2 text-sm"><div class="w-2 h-2 rounded-full bg-purple-500"></div><span class="font-medium">Longer Battery Life</span><span class="text-xs text-muted-foreground">â Tie-breaker preference</span></div></div></div></div></div></div></div></div><div class="flex flex-col md:flex-row gap-6 items-start"><div class="flex-shrink-0 w-16 h-16 rounded-2xl bg-primary text-primary-foreground flex items-center justify-center text-2xl font-bold">2</div><div class="flex-1"><div class="flex items-start gap-3 mb-3"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-filter h-6 w-6 text-primary flex-shrink-0 mt-1"><polygon points="22 3 2 3 10 12.46 10 19 14 21 14 12.46 22 3"></polygon></svg><h3 class="text-2xl font-bold">Filter by constraints (fast)</h3></div><p class="text-muted-foreground mb-4">Eliminate candidates that fail any hard constraint. Use indexed lookups, bitmap filters, or Boolean formulas for millisecond filtering even on thousands of products.</p><div class="rounded-2xl border bg-card text-card-foreground shadow-lg"><div class="flex flex-col space-y-1.5 p-6"><h3 class="font-semibold tracking-tight text-lg">Example: From 3,000 to 47</h3></div><div class="p-6 pt-0 space-y-2 text-sm text-muted-foreground"><p>⢠Started with 3,000 laptops</p><p>⢠Failed RAM requirement: 1,200 eliminated</p><p>⢠Failed SSD requirement: 800 eliminated</p><p>⢠Failed weight threshold: 700 eliminated</p><p>⢠Failed seller rating: 253 eliminated</p><p class="font-semibold text-foreground pt-2">â 47 candidates remain for scoring</p></div></div></div></div><div class="flex flex-col md:flex-row gap-6 items-start"><div class="flex-shrink-0 w-16 h-16 rounded-2xl bg-primary text-primary-foreground flex items-center justify-center text-2xl font-bold">3</div><div class="flex-1"><div class="flex items-start gap-3 mb-3"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-git-branch h-6
1w-6 text-primary flex-shrink-0 mt-1"><line x1="6" x2="6" y1="3" y2="15"></line><circle cx="18" cy="6" r="3"></circle><circle cx="6" cy="18" r="3"></circle><path d="M18 9a9 9 0 0 1-9 9"></path></svg><h3 class="text-2xl font-bold">Build a DAG (partial order) with dominance weights</h3></div><p class="text-muted-foreground mb-4">Organize criteria into levels where higher-priority wins can't be gamed by stacking lower-level features. Assign level-dominant weights or use lexicographic vectors.</p><div class="rounded-2xl border bg-card text-card-foreground shadow-lg"><div class="flex flex-col space-y-1.5 p-6"><h3 class="font-semibold tracking-tight text-lg">How the DAG Creates Dominance</h3><p class="text-sm text-muted-foreground">The partial order determines which laptops are optimal</p></div><div class="p-6 pt-0 space-y-4 text-sm"><div class="p-4 bg-gradient-to-r from-blue-50 to-indigo-50 dark:from-blue-950 dark:to-indigo-950 rounded-lg border-l-4 border-blue-500"><h4 class="font-semibold text-foreground mb-2">Preference Comparisons</h4><p class="text-muted-foreground mb-3">The system evaluates each laptop against the partial order preferences. A laptop satisfying more preference comparisons dominates one satisfying fewer.</p><div class="space-y-2 text-xs"><div class="flex items-start gap-2"><span class="text-green-600 dark:text-green-400">â</span><span><strong>Gray</strong> is preferred over Black (meets 'gray > black')</span></div><div class="flex items-start gap-2"><span class="text-green-600 dark:text-green-400">â</span><span><strong>Intel</strong> is preferred over AMD (meets 'intel > amd')</span></div><div class="flex items-start gap-2"><span class="text-green-600 dark:text-green-400">â</span><span><strong>New</strong> is preferred over Refurbished (meets 'new > refurbished')</span></div><div class="flex items-start gap-2"><span class="text-purple-600 dark:text-purple-400">â</span><span class="text-purple-700 dark:text-purple-300">Implicit preferences (rating, battery, etc.) only used to break ties</span></div></div></div><div class="p-4 bg-green-50 dark:bg-green-950 rounded-lg border-l-4 border-green-500"><h4 class="font-semibold text-foreground mb-2">â Gaming-Resistant Property</h4><p class="text-muted-foreground">A laptop matching more DAG preferences (Gray + Intel + New = 3/3 preferences met) will <strong>always dominate</strong>one matching fewer preferences (Black + Intel + New = 2/3 preferences met), even if the latter excels on implicit factors like price or battery life.<span class="block mt-2 text-xs italic">The partial order ensures explicit preferences cannot be overridden by implicit tie-breakers.</span></p></div><div class="p-4 bg-amber-50 dark:bg-amber-950 rounded-lg border-l-4 border-amber-500"><h4 class="font-semibold text-foreground mb-2">How It Differs From Traditional Ranking</h4><p class="text-muted-foreground text-xs">Traditional ML ranking might weight features equally: Price (30%), Battery (25%), Rating (20%), Color (15%), Processor (10%). A cheap laptop with great battery/rating could beat your preferred Gray+Intel choice. With dominance-based DAG, explicit preferences (color, processor, condition) <strong>always</strong> dominate implicit tie-breakers.</p></div></div></div></div></div><div class="flex flex-col md:flex-row gap-6 items-start"><div class="flex-shrink-0 w-16 h-16 rounded-2xl bg-primary text-primary-foreground flex items-center justify-center text-2xl font-bold">4</div><div class="flex-1"><div class="flex items-start gap-3 mb-3"><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-target h-6 w-6 text-primary flex-shrink-0 mt-1"><circle cx="12" cy="12" r="10"></circle><circle cx="12" cy="12" r="6"></circle><circle cx="12" cy="12" r="2"></circle></svg><h3 class="text-2xl font-bold">Score survivors; return optimal or co-optimal results</h3></div><p class="text-muted-foreground mb-4">Apply the preference DAG to rank surviving candidates. Return the single optimal choice, or all c
1o-optimal results (if multiple items tie at the top) with a level-by-level explanation.</p><div class="space-y-4"><div class="rounded-2xl bg-card text-card-foreground shadow-lg border-2 border-green-500"><div class="flex flex-col space-y-1.5 p-6 bg-green-50 dark:bg-green-950"><div class="flex items-center gap-2"><span class="text-2xl">ð</span><h3 class="font-semibold tracking-tight text-lg">Optimal Result</h3></div><p class="text-sm text-muted-foreground">Dell XPS 13 Plus - Gray, Intel i7-1360P, New</p></div><div class="p-6 space-y-3 pt-4"><div class="grid grid-cols-2 gap-3 text-sm mb-4"><div><strong>Price:</strong> $949</div><div><strong>RAM:</strong> 16GB</div><div><strong>Storage:</strong> 512GB SSD</div><div><strong>Weight:</strong> 1.24kg</div><div><strong>USB-C:</strong> 2 ports</div><div><strong>Wi-Fi:</strong> 6E</div><div><strong>Battery:</strong> 12h</div><div><strong>Rating:</strong> 4.7/5</div></div><div class="p-3 border-l-4 border-blue-500 bg-blue-50 dark:bg-blue-950 rounded text-sm"><p class="font-semibold text-foreground mb-1">â Mandatory constraints:</p><p class="text-muted-foreground">Passed all: priceâ¤$1000, RAMâ¥16GB, storageâ¥512GB, weightâ¤1.5kg, 2ÃUSB-C, Wi-Fi 6E, delivers by Nov 1</p></div><div class="p-3 border-l-4 border-primary bg-primary/5 rounded text-sm"><p class="font-semibold text-foreground mb-1">â DAG Preferences:</p><div class="text-muted-foreground space-y-1 mt-2"><div class="flex items-center gap-2"><span class="text-green-600 dark:text-green-400">â</span><span>Gray color (meets 'gray > black')</span></div><div class="flex items-center gap-2"><span class="text-green-600 dark:text-green-400">â</span><span>Intel i7-1360P (meets 'intel > amd')</span></div><div class="flex items-center gap-2"><span class="text-green-600 dark:text-green-400">â</span><span>New condition (meets 'new > refurbished')</span></div></div></div><div class="p-3 border-l-4 border-green-500 bg-green-50 dark:bg-green-950 rounded text-sm"><p class="font-semibold text-green-900 dark:text-green-100 mb-1">ð Why optimal:</p><p class="text-green-800 dark:text-green-200">This laptop satisfies all mandatory constraints AND all three DAG preference comparisons (3/3 met). No other candidate dominates it on the partial order. It represents the unique optimal choice.</p></div></div></div><div class="rounded-2xl bg-card text-card-foreground shadow-lg border-2 border-orange-300"><div class="flex flex-col space-y-1.5 p-6 bg-orange-50 dark:bg-orange-950"><div class="flex items-center gap-2"><span class="text-2xl">â ï¸</span><h3 class="font-semibold tracking-tight text-lg">Non-Optimal Result #1</h3></div><p class="text-sm text-muted-foreground">HP Spectre x360 - Black, Intel i7-1355U, New</p></div><div class="p-6 space-y-3 pt-4"><div class="grid grid-cols-2 gap-3 text-sm mb-4"><div><strong>Price:</strong> $929</div><div><strong>RAM:</strong> 16GB</div><div><strong>Storage:</strong> 512GB SSD</div><div><strong>Weight:</strong> 1.36kg</div><div><strong>USB-C:</strong> 2 ports</div><div><strong>Wi-Fi:</strong> 6E</div><div><strong>Battery:</strong> 14h</div><div><strong>Rating:</strong> 4.8/5</div></div><div class="p-3 border-l-4 border-blue-500 bg-blue-50 dark:bg-blue-950 rounded text-sm"><p class="font-semibold text-foreground mb-1">â Mandatory constraints:</p><p class="text-muted-foreground">Passed all constraints</p></div><div class="p-3 border-l-4 border-primary bg-primary/5 rounded text-sm"><p class="font-semibold text-foreground mb-1">DAG Preferences:</p><div class="text-muted-foreground space-y-1 mt-2"><div class="flex items-center gap-2"><span class="text-red-600 dark:text-red-400">â</span><span>Black color (fails 'gray > black')</span></div><div class="flex items-center gap-2"><span class="text-green-600 dark:text-green-400">â</span><span>Intel i7-1355U (meets 'intel > amd')</span></div><div class="flex items-center gap-2"><span class="text-green-600 dark:text-green-400">â</span><span>New condition (meets 'new > refurbished')</span></div></div></div><div class="p-3 border-l-4 border-orange-500 bg-orange-50 dark:bg-orange-950 rounded text-sm"><p class="font-semibold text-orange-900 dark:text-orange-100 mb-1">Why NOT optimal:</p><p class="text-orange-800 dark:text-orange-200">Even though this laptop is cheaper ($929 vs $949) and has better battery life (14h vs 12h), it's <strong>dominated</strong> by the optimal choice because it meets only 2/3 DAG preferences (black instead of gray). Implicit advantages (price, battery) cannot overcome explicit preference differences in the partial order.</p></div></div></div><div class="rounded-2xl bg-card text-card-foreground shadow-lg border-2 border-red-300"><div class="flex flex-col space-y-1.5 p-6 bg-red-50 dark:bg-red-950"><div class="flex items-center gap-2"><span class="text-2xl">â</span>
1<h3 class="font-semibold tracking-tight text-lg">Non-Optimal Result #2</h3></div><p class="text-sm text-muted-foreground">Lenovo ThinkPad X1 - Gray, AMD Ryzen 7, New</p></div><div class="p-6 space-y-3 pt-4"><div class="grid grid-cols-2 gap-3 text-sm mb-4"><div><strong>Price:</strong> $899</div><div><strong>RAM:</strong> 16GB</div><div><strong>Storage:</strong> 512GB SSD</div><div><strong>Weight:</strong> 1.19kg</div><div><strong>USB-C:</strong> 2 ports</div><div><strong>Wi-Fi:</strong> 6E</div><div><strong>Battery:</strong> 15h</div><div><strong>Rating:</strong> 4.9/5</div></div><div class="p-3 border-l-4 border-blue-500 bg-blue-50 dark:bg-blue-950 rounded text-sm"><p class="font-semibold text-foreground mb-1">â Mandatory constraints:</p><p class="text-muted-foreground">Passed all constraints</p></div><div class="p-3 border-l-4 border-primary bg-primary/5 rounded text-sm"><p class="font-semibold text-foreground mb-1">DAG Preferences:</p><div class="text-muted-foreground space-y-1 mt-2"><div class="flex items-center gap-2"><span class="text-green-600 dark:text-green-400">â</span><span>Gray color (meets 'gray > black')</span></div><div class="flex items-center gap-2"><span class="text-red-600 dark:text-red-400">â</span><span>AMD Ryzen 7 (fails 'intel > amd')</span></div><div class="flex items-center gap-2"><span class="text-green-600 dark:text-green-400">â</span><span>New condition (meets 'new > refurbished')</span></div></div></div><div class="p-3 border-l-4 border-red-500 bg-red-50 dark:bg-red-950 rounded text-sm"><p class="font-semibold text-red-900 dark:text-red-100 mb-1">Why NOT optimal:</p><p class="text-red-800 dark:text-red-200">Despite being cheaper ($899), lighter (1.19kg), having better battery (15h), and higher rating (4.9/5), this laptop is <strong>dominated</strong> by the optimal choice because it meets only 2/3 DAG preferences (AMD instead of Intel). Gaming-resistant: you can't "win" by stacking implicit features (price, weight, battery) if you fail explicit DAG preferences.</p></div></div></div><div class="rounded-2xl bg-card text-card-foreground shadow-lg border-2 border-gray-400"><div class="flex flex-col space-y-1.5 p-6 bg-gray-50 dark:bg-gray-900"><div class="flex items-center gap-2"><span class="text-2xl">ð«</span><h3 class="font-semibold tracking-tight text-lg">Excluded Result</h3></div><p class="text-sm text-muted-foreground">ASUS ZenBook - Silver, Intel i7-1355U, New</p></div><div class="p-6 space-y-3 pt-4"><div class="grid grid-cols-2 gap-3 text-sm mb-4"><div><strong>Price:</strong> $1,099</div><div><strong>RAM:</strong> 16GB</div><div><strong>Storage:</strong> 512GB SSD</div><div><strong>Weight:</strong> 1.29kg</div><div><strong>USB-C:</strong> 2 ports</div><div><strong>Wi-Fi:</strong> 6E</div><div><strong>Battery:</strong> 13h</div><div><strong>Rating:</strong> 4.6/5</div></div><div class="p-3 border-l-4 border-red-500 bg-red-50 dark:bg-red-950 rounded text-sm"><p class="font-semibold text-foreground mb-1">â Mandatory constraints:</p><p class="text-muted-foreground">Failed: price=$1,099 > $1,000 maximum</p></div><div class="p-3 border-l-4 border-gray-500 bg-gray-50 dark:bg-gray-900 rounded text-sm"><p class="font-semibold text-gray-900 dark:text-gray-100 mb-1">Why EXCLUDED:</p><p class="text-gray-800 dark:text-gray-200">This laptop violates a hard constraint (price > $1,000), so it's eliminated before preference evaluation. Hard constraints are non-negotiable filters applied first.</p></div></div></div></div></div></div></div></div></section><section class="py-16 sm:py-24 bg-muted/30"><div class="mx-auto max-w-7xl px-6 lg:px-8"><div class="mx-auto max-w-2xl text-center mb-12"><h2 class="text-3xl font-bold tracking-tight sm:text-4xl mb-4">Why This Is Not a Trivial Problem</h2><p class="text-lg text-muted-foreground">Finding optimal solutions with preferences requires solving fundamental computational challenges</p></div><div class="mx-auto max-w-5xl space-y-8"><div class="rounded-2xl border bg-card text-card-foreground shadow-lg border-l-4 border-red-500"><div class="flex flex-col space-y-1.5 p-6"><h3 class="font-semibold tracking-tight text-2xl">
1Traditional Approaches Require Expensive Dominance Testing</h3><p class="text-sm text-muted-foreground">Without our invention, systems must compare every solution against all others</p></div><div class="p-6 pt-0 space-y-4"><p class="text-muted-foreground">When a system "sees" a potential solution, traditional approaches <strong>cannot</strong> determine if it's optimal without comparing it against <strong>all previously found solutions</strong> via computationally expensive dominance testing procedures.</p><div class="bg-red-50 dark:bg-red-950 p-4 rounded-lg border border-red-200 dark:border-red-800"><h4 class="font-semibold text-red-900 dark:text-red-100 mb-2">The Dominance Testing Problem:</h4><ul class="space-y-2 text-sm text-red-800 dark:text-red-200"><li>⢠For <strong>each solution</strong>, perform a dominance test with <strong>all other solutions</strong></li><li>⢠Computational cost grows <strong>quadratically (O(N²))</strong> with the number of solutions</li><li>⢠With 1,000 solutions: ~1,000,000 comparisons needed</li><li>⢠With 10,000 solutions: ~100,000,000 comparisons needed</li></ul></div><p class="text-muted-foreground"><strong>Worse still:</strong> Such approaches require that <strong>all solutions are already available</strong> before starting to calculate optimal results. This is <strong>impractical and infeasible</strong> in real-world, industrial use cases dealing with:</p><div class="grid md:grid-cols-3 gap-4"><div class="p-3 bg-muted rounded-lg"><div class="font-semibold mb-1">Huge Volumes</div><div class="text-xs text-muted-foreground">Millions of products from multiple sources</div></div><div class="p-3 bg-muted rounded-lg"><div class="font-semibold mb-1">Progressive Arrival</div><div class="text-xs text-muted-foreground">Solutions appear incrementally over time</div></div><div class="p-3 bg-muted rounded-lg"><div class="font-semibold mb-1">Multiple Data Sources</div><div class="text-xs text-muted-foreground">Parallel queries to different databases/APIs</div></div></div></div></div><div class="rounded-2xl border bg-card text-card-foreground shadow-lg border-l-4 border-amber-500"><div class="flex flex-col space-y-1.5 p-6"><h3 class="font-semibold tracking-tight text-2xl">Trivial Scoring Methods Are Mathematically Wrong</h3><p class="text-sm text-muted-foreground">Simple linear scoring leads to falsely "optimal" solutions</p></div><div class="p-6 pt-0 space-y-4"><p class="text-muted-foreground">A naive approach might try to compute weights for DAG nodes by simply summing them up following the partial order, or using linearly increasing weights from top to bottom preferences. <strong>This is mathematically incorrect</strong> and leads to falsely "optimal" solutions.</p><div class="bg-amber-50 dark:bg-amber-950 p-4 rounded-lg border border-amber-200 dark:border-amber-800"><h4 class="font-semibold text-amber-900 dark:text-amber-100 mb-3">
1Example: Why Linear Weights Fail</h4><p class="text-sm text-amber-800 dark:text-amber-200 mb-3">Consider a DAG with preferences at different levels. A trivial system using simple additive or linear weights will produce incorrect results where lower-level preferences can overwhelm higher-level ones.</p><div class="bg-red-100 dark:bg-red-900 p-3 rounded text-sm"><strong class="text-red-900 dark:text-red-100">The Problem:</strong><p class="text-red-800 dark:text-red-200 mt-1">With linear weighting, a solution satisfying multiple lower-priority preferences can incorrectly outscore a solution satisfying a single higher-priority preference. <strong>This violates the dominance hierarchy and produces falsely "optimal" solutions.</strong></p></div></div><p class="text-sm text-muted-foreground italic">This trivial approach only works correctly in the special case of a simple linear chain of preferences, but fails for general DAGs with multiple roots and parallel branches.</p></div></div></div></div></section><section class="py-16 sm:py-24"><div class="mx-auto max-w-7xl px-6 lg:px-8"><div class="mx-auto max-w-2xl text-center mb-12"><h2 class="text-3xl font-bold tracking-tight sm:text-4xl mb-4">How Our Invention Solves These Problems</h2><p class="text-lg text-muted-foreground">Our patent-pending method enables progressive scoring without dominance testing</p></div><div class="mx-auto max-w-5xl space-y-8"><div class="rounded-2xl text-card-foreground shadow-lg border-2 border-green-500 bg-green-50/30 dark:bg-green-950/20"><div class="flex flex-col space-y-1.5 p-6"><h3 class="font-semibold tracking-tight text-2xl">Key Innovation: Direct Optimality Scoring</h3><p class="text-sm text-muted-foreground">Score each solution independently using only the DAG structure</p></div><div class="p-6 pt-0 space-y-6"><p class="text-muted-foreground">When the system "sees" a potential solution, it can produce an <strong>optimality score calculated using only the current solution and the DAG</strong> â no comparison with other solutions needed!</p><div><h4 class="font-bold text-lg mb-3">â
Properties Guaranteed by Our System:</h4><div class="grid md:grid-cols-2 gap-4"><div class="bg-blue-50 dark:bg-blue-950 p-4 rounded-lg border border-blue-200 dark:border-blue-800"><div class="font-semibold text-blue-900 dark:text-blue-100 mb-2">ð Pareto-Optimality</div><p class="text-sm text-blue-800 dark:text-blue-200">Solutions with maximum score are guaranteed to be Pareto-optimal with respect to the DAG preferences. No other solution dominates them.</p></div><div class="bg-purple-50 dark:bg-purple-950 p-4 rounded-lg border border-purple-200 dark:border-purple-800"><div class="font-semibold text-purple-900 dark:text-purple-100 mb-2">âï¸ Equivalence Property</div><p class="text-sm text-purple-800 dark:text-purple-200">Solutions with equal scores are truly equivalent with respect to preferences â they satisfy the same preference comparisons.</p></div><div class="bg-green-50 dark:bg-green-950 p-4 rounded-lg border border-green-200 dark:border-green-800"><div class="font-semibold text-green-900 dark:text-green-100 mb-2">â¡ Progressive Computation</div><p class="text-sm text-green-800 dark:text-green-200">Solutions can arrive progressively (from multiple data sources, streaming APIs, etc.) and be scored immediately without waiting for all solutions.</p></div><div class="bg-orange-50 dark:bg-orange-950 p-4 rounded-lg border border-orange-200 dark:border-orange-800"><div class="font-semibold text-orange-900 dark:text-orange-100 mb-2">â»ï¸ No Recalculation Needed</div><p class="text-sm text-orange-800 dark:text-orange-200">When new solutions arrive, their scores remain valid. No need to recalculate or restart the computation from scratch.</p></div></div></div><div class="bg-gradient-to-r from-cyan-50 to-blue-50 dark:from-cyan-950 dark:to-blue-950 p-5 rounded-lg border-2 border-cyan-500"><h4 class="font-bold text-lg mb-3 text-cyan-900 dark:text-cyan-100">ð Dramatic Computational Improvement</h4><div class="grid md:grid-cols-2 gap-4 mb-4"><div class="bg-red-100 dark:bg-red-900 p-4 rounded border-2 border-red-400"><div class="font-semibold text-red-900 dark:text-red-100 mb-2">
1â Traditional Approach</div><div class="font-mono text-xl text-red-700 dark:text-red-300 mb-2">O(N² à V)</div><p class="text-xs text-red-800 dark:text-red-200">Quadratic growth: must compare each solution against all others (N²) for each preference criterion (V)</p></div><div class="bg-green-100 dark:bg-green-900 p-4 rounded border-2 border-green-400"><div class="font-semibold text-green-900 dark:text-green-100 mb-2">â
Our System</div><div class="font-mono text-xl text-green-700 dark:text-green-300 mb-2">O(N à V + N log N)</div><p class="text-xs text-green-800 dark:text-green-200">Near-linear growth: score each solution once (N à V), then sort results (N log N)</p></div></div><div class="bg-white dark:bg-gray-900 p-4 rounded"><h5 class="font-semibold mb-2 text-sm">Real-World Impact:</h5><div class="space-y-2 text-xs text-muted-foreground"><div class="flex justify-between items-center"><span>1,000 solutions, 10 preferences:</span><span class="font-semibold"><span class="text-red-600">10M ops</span> â <span class="text-green-600">~10K ops</span><span class="text-green-600 ml-2">(1,000à faster)</span></span></div><div class="flex justify-between items-center"><span>10,000 solutions, 20 preferences:</span><span class="font-semibold"><span class="text-red-600">2B ops</span> â <span class="text-green-600">~200K ops</span><span class="text-green-600 ml-2">(10,000à faster)</span></span></div><div class="flex justify-between items-center"><span>100,000 solutions, 50 preferences:</span><span class="font-semibold"><span class="text-red-600">500B ops</span> â <span class="text-green-600">~5.8M ops</span><span class="text-green-600 ml-2">(86,000à faster)</span></span></div></div></div></div><div class="bg-indigo-50 dark:bg-indigo-950 p-4 rounded-lg border border-indigo-200 dark:border-indigo-800"><h4 class="font-bold text-lg mb-3 text-indigo-900 dark:text-indigo-100">âï¸ Parallel & Distributed Processing</h4><p class="text-sm text-indigo-800 dark:text-indigo-200 mb-3">Because each solution's optimality score is independent and stable, our system supports:</p><ul class="space-y-2 text-sm text-indigo-800 dark:text-indigo-200"><li>⢠<strong>Parallel scoring</strong> of solutions across multiple CPU cores</li><li>⢠<strong>Distributed computation</strong> using "partial rankers" for solution subsets</li><li>⢠<strong>Incremental updates</strong> as new solutions arrive from different data sources</li><li>⢠<strong>Real-time ranking</strong> without reprocessing the entire solution set</li></ul></div><div class="bg-gradient-to-r from-green-500 to-emerald-500 text-white p-6 rounded-lg text-center"><div class="text-4xl font-bold mb-2">Millisecond Response Times</div><p class="text-green-100">Find optimal solutions from thousands of candidates in real-time, enabling live search, auto-suggest, and interactive user experiences</p></div></div></div></div></div></section><section class="py-16 sm:py-24 bg-muted/30"><div class="mx-auto max-w-7xl px-6 lg:px-8"><div class="mx-auto max-w-2xl text-center mb-12"><h2 class="text-3xl font-bold tracking-tight sm:text-4xl">Key properties</h2></div><div class="mx-auto max-w-5xl grid grid-cols-1 md:grid-cols-2 gap-6"><div class="rounded-2xl border bg-card text-card-foreground shadow-lg"><div class="flex flex-col space-y-1.5 p-6"><h3 class="text-2xl font-semibold leading-none tracking-tight">Provably optimal</h3></div><div class="p-6 pt-0"><p class="text-sm text-muted-foreground">Dominance weights ensure higher-level wins can't be overturned by gaming lower levels. Patent-backed methodology (US 2025/0298806 A1).</p></div></div><div class="rounded-2xl border bg-card text-card-foreground shadow-lg"><div class="flex flex-col space-y-1.5 p-6"><h3 class="text-2xl font-semibold leading-none tracking-tight">Millisecond performance</h3></div><div class="p-6 pt-0"><p class="text-sm text-muted-foreground">Constraint filters and efficient scoring return results in real-time. Fast enough for live search, auto-suggest, and A/B testing.</p></div></div><div class="rounded-2xl border bg-card text-card-foreground shadow-lg"><div class="flex flex-col space-y-1.5 p-6"><h3 class="text-2xl font-semibold leading-none tracking-tight">Explainable</h3></div><div class="p-6 pt-0"><p class="text-sm text-muted-foreground">Every result includes a level-by-level rationale. Users and auditors can see exactly why each option won or lost.</p></div></div><div class="rounded-2xl border bg-card text-card-foreground shadow-lg"><div class="flex flex-col space-y-1.5 p-6"><h3 class="text-2xl font-semibold leading-none tracking-tight">Gaming-resistant</h3></div><div class="p-6 pt-0"><p class="text-sm text-muted-foreground">Sellers can't boost rank by adding irrelevant features. Only improvements at the right dominance level matter.</p></div></div></div></div></section><section class="py-16 sm:py-24"><div class="mx-auto max-w-7xl px-6 lg:px-8"><div class="text-center mb-10"><h2 class="text-3xl font-bold mb-4">Comparison with State-of-the-Art Methods</h2><p class="text-lg text-muted-foreground max-w-3xl mx-auto">See how our Preference-DAG Optimality Scoring compares to existing approaches across key dimensions.</p></div><div class="overflow-x-auto mb-8"><table class="w-full border-collapse text-sm"><thead><tr class="bg-muted"><th class="border border-border p-3 text-left font-semibold min-w-[180px]">Approach</th><th class="border border-border p-3 text-left font-semibold min-w-[120px]">Fast at Web Scale?</th><th class="border border-border p-3 text-left font-semibold min-w-[120px]">Gaming-Resistant?</th><th class="border border-border p-3 text-left font-semibold min-w-[120px]">Explainable?</th><th class="border border-border p-3 text-left font-semibold min-w-[140px]">Provably Optimal?</th><th class="border border-border p-3 text-left font-semibold min-w-[160px]">Streaming-Compatible?<sup class="text-[10px]">1</sup></th><th class="border border-border p-3 text-left font-semibold min-w-[200px]">Algorithmic Complexity<sup class="text-[10px]">2</sup></th></tr></thead><tbody><tr><td class="border border-border p-3 font-medium">Traditional ML Ranking<br/><span class="text-xs text-muted-foreground">Google, Amazon, Bing</span></td><td class="border border-border p-3"><span class="text-green-600">â
Yes</span></td><td class="border border-border p-3"><span class="text-amber-600">â ï¸ SEO arms race</span></td><td class="border border-border p-3"><span class="text-amber-600">â ï¸ Opaque</span></td><td class="border border-border p-3"><span class="text-red-600">â No proofs</span></td><td class="border border-border p-3"><span class="text-amber-600">â ï¸ System-level only</span></td><td class="border border-border p-3 text-xs">O(N log N)<br/><span class="text-muted-foreground">Fixed candidate set</span></td></tr><tr class="bg-muted/50"><td class="border border-border p-3 font-medium">Recommendation Engines<br/><span class="text-xs text-muted-foreground">Netflix, Spotify, Amazon Recs</span></td><td class="border border-border p-3"><span class="text-green-600">â
Yes</span></td><td class="border border-border p-3"><span class="text-amber-600">â ï¸ Fraud detection</span></td><td class="border border-border p-3"><span class="text-amber-600">â ï¸ Black-box</span></td><td class="border border-border p-3"><span class="text-red-600">â Engagement proxy</span></td><td class="border border-border p-3"><span class="text-amber-600">â ï¸ Session-level only</span></td><td class="border border-border p-3 text-xs">O(N)<br/><span class="text-muted-foreground">Retrieval, not optimality</span></td></tr><tr><td class="border border-border p-3 font-medium">MCDA / Pareto / Skyline<br/><span class="text-xs text-muted-foreground">AHP, TOPSIS, ELECTRE</span></td><td class="border border-border p-3"><span class="text-amber-600">â ï¸ Limited scale</span></td><td class="border border-border p-3"><span class="text-amber-600">â ï¸ Gameable</span></td><td class="border border-border p-3"><span class="text-green-600">â
Yes</span></td><td class="border border-border p-3"><span class="text-green-600">â
Pareto-optimal</span></td><td class="border border-border p-3"><span class="text-red-600">â No</span></td><td class="border border-border p-3 text-xs">O(N²ÃV)<br/><span class="text-muted-foreground">Dominance-testing based</span></td></tr><tr class="bg-muted/50"><td class="border border-border p-3 font-medium">LLM-Enhanced Search<br/><span class="text-xs text-muted-foreground">Perplexity, ChatGPT+Search</span></td><td class="border border-border p-3"><span class="text-amber-600">â ï¸ Slower (seconds)</span></td><td class="border border-border p-3"><span class="text-amber-600">â ï¸ Prompt injection</span></td><td class="border border-border p-3"><span class="text-amber-600">â ï¸ Citations only</span></td><td class="border border-border p-3"><span class="text-red-600">â No guarantees</span></td><td class="border border-border p-3"><span class="text-amber-600">â ï¸ Conversational only</span></td><td class="border border-border p-3 text-xs">O(N log N + T)<br/><span class="text-muted-foreground">Retrieval + generation</span></td></tr><tr><td class="border border-border p-3 font-medium">Generic Preference-Based<br/><span class="text-xs text-muted-foreground">Pareto, CP-Net, Utility</span></td><td class="border border-border p-3"><span class="text-amber-600">â ï¸ Moderate</span></td><td class="border border-border p-3"><span class="text-amber-600">â ï¸ Data gameable</span></td><td class="border border-border p-3"><span class="text-green-600">â
Yes</span></td><td class="border border-border p-3"><span class="text-green-600">â
Under model</span></td><td class="border border-border p-3"><span class="text-red-600">â No</span></td><td class="border border-border p-3 text-xs">NP-hard<br/><span class="text-muted-foreground">CP-net dominance testing</span></td></tr><tr class="bg-primary/10 border-2 border-primary"><td class="border border-primary p-3 font-bold text-primary">Preference-DAG (US 2025/0298806)<br/><span class="text-xs font-normal">PerfectFinder</span></td><td class="border border-primary p-3"><span class="text-green-600 font-semibold">â
Yes</span></td><td class="border border-primary p-3"><span class="text-green-600 font-semibold">â
Yes</span></td><td class="border border-primary p-3"><span class="text-green-600 font-semibold">â
Full audit</span></td><td class="border border-primary p-3"><span class="text-green-600 font-semibold">â
Guaranteed</span></td><td class="border border-primary p-3"><span class="text-green-600 font-semibold">â
Full streaming</span></td><td class="border border-primary p-3 text-xs font-semibold">O(NÃV + N log N)<br/><span class="font-normal text-muted-foreground">No dominance testing</span></td></tr></tbody></table></div><div class="bg-muted/50 p-4 rounded-lg text-sm space-y-4"><div><p class="font-semibold mb-2">Notation:</p><div class="grid grid-cols-2 md:grid-cols-3 gap-x-6 gap-y-1 text-muted-foreground text-xs"><div><strong>N</strong> = number of candidate items</div><div><strong>V</strong> = number of DAG nodes (preferences)</div><div><strong>E</strong> = number of DAG edges</div><div><strong>T</strong> = tokens in LLM context</div></div></div><div class="border-t border-border pt-3 space-y-2 text-xs text-muted-foreground"><p><sup>1</sup> <strong>Streaming-Compatible:</strong> Can solutions be evaluated and ranked as they arrive from multiple sources, without requiring the full solution set upfront? â
= algorithmically progressive; â ï¸ = UI/system-level only; â = requires full dataset.</p><p><sup>2</sup> <strong>Algorithmic Complexity:</strong> Worst-case per-query cost (Big O notation) for computing optimal ranking, excluding offline training, index building, and caching.</p></div></div><div class="bg-gradient-to-r from-blue-50 to-indigo-50 dark:from-blue-950 dar
1k:to-indigo-950 p-6 rounded-lg border border-blue-200 dark:border-blue-800 mt-8"><h3 class="text-lg font-bold mb-4">Practical Example: What These Numbers Mean</h3><div class="grid md:grid-cols-2 gap-6"><div><p class="text-sm font-semibold mb-2">Scenario:</p><ul class="text-sm text-muted-foreground space-y-1"><li>⢠<strong>V = 10</strong> preference nodes (e.g., color, brand, price range...)</li><li>⢠<strong>E = 15</strong> edges in the preference DAG</li><li>⢠<strong>N = 10,000</strong> candidate products (streaming from multiple sources)</li><li>⢠CPU speed: ~1 billion simple operations/second</li></ul></div><div><p class="text-sm font-semibold mb-2">Computation Time:</p><div class="overflow-x-auto"><table class="text-xs w-full"><thead><tr class="border-b border-blue-200 dark:border-blue-700"><th class="text-left py-1 pr-3">Method</th><th class="text-left py-1 pr-3">Operations</th><th class="text-left py-1">Time</th></tr></thead><tbody class="text-muted-foreground"><tr><td class="py-1 pr-3">MCDA/Pareto O(N²ÃV)</td><td class="py-1 pr-3">10,000² à 10 = 1B</td><td class="py-1 text-red-600 font-semibold">~1 second</td></tr><tr><td class="py-1 pr-3">CP-Net (NP-hard)</td><td class="py-1 pr-3">Exponential</td><td class="py-1 text-red-600 font-semibold">secondsâminutes</td></tr><tr class="bg-green-100 dark:bg-green-900/30"><td class="py-1 pr-3 font-semibold">PerfectFinder O(NÃV + N log N)</td><td class="py-1 pr-3">100K + 133K â 233K</td><td class="py-1 text-green-600 font-semibold">~0.2 ms</td></tr></tbody></table></div><p class="text-xs text-muted-foreground mt-2 italic">PerfectFinder is <strong>~4,300à faster</strong> than dominance-testing approaches for this scenario.</p></div></div></div></div></section><section class="bg-muted/30 py-16 sm:py-24"><div class="mx-auto max-w-7xl px-6 lg:px-8 text-center"><h2 class="text-3xl font-bold tracking-tight sm:text-4xl mb-6">See it in action</h2><p class="text-lg leading-8 text-muted-foreground max-w-2xl mx-auto mb-10">Try our interactive demos or explore use cases for your industry</p><div class="flex items-center justify-center gap-x-6"><a href="/demos"><button class="inline-flex items-center justify-center whitespace-nowrap font-medium ring-offset-background transition-all focus-visible:outline-none focus-visible:ring-2 focus-visible:ring-ring focus-visible:ring-offset-2 disabled:pointer-events-none disabled:opacity-50 bg-gradient-to-r from-indigo-500 via-purple-500 to-purple-400 text-white hover:shadow-xl hover:shadow-purple-500/50 hover:scale-105 shadow-lg h-14 roun
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1:24,\"viewBox\":\"0 0 24 24\",\"fill\":\"none\",\"stroke\":\"currentColor\",\"strokeWidth\":2,\"strokeLinecap\":\"round\",\"strokeLinejoin\":\"round\",\"className\":\"lucide lucide-filter h-6 w-6 text-primary flex-shrink-0 mt-1\",\"children\":[[\"$\",\"polygon\",\"1yg77f\",{\"points\":\"22 3 2 3 10 12.46 10 19 14 21 14 12.46 22 3\"}],\"$undefined\"]}],[\"$\",\"h3\",null,{\"className\":\"text-2xl font-bold\",\"children\":\"Filter by constraints (fast)\"}]]}],[\"$\",\"p\",null,{\"className\":\"text-muted-foreground mb-4\",\"children\":\"Eliminate candidates that fail any hard constraint. Use indexed lookups, bitmap filters, or Boolean formulas for millisecond filtering even on thousands of products.\"}],[\"$\",\"div\",null,{\"className\":\"rounded-2xl border bg-card text-card-foreground shadow-lg\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex flex-col space-y-1.5 p-6\",\"children\":[\"$\",\"h3\",null,{\"className\":\"font-semibold tracking-tight text-lg\",\"children\":\"Example: From 3,000 to 47\"}]}],[\"$\",\"div\",null,{\"className\":\"p-6 pt-0 space-y-2 text-sm text-muted-foreground\",\"children\":[[\"$\",\"p\",null,{\"children\":\"⢠Started with 3,000 laptops\"}],[\"$\",\"p\",null,{\"children\":\"⢠Failed RAM requirement: 1,200 eliminated\"}],[\"$\",\"p\",null,{\"children\":\"⢠Failed SSD requirement: 800 eliminated\"}],[\"$\",\"p\",null,{\"children\":\"⢠Failed weight threshold: 700 eliminated\"}],[\"$\",\"p\",null,{\"children\":\"⢠Failed seller rating: 253 eliminated\"}],[\"$\",\"p\",null,{\"className\":\"font-semibold text-foreground pt-2\",\"children\":\"â 47 candidates remain for scoring\"}]]}]]}]]}]]}],[\"$\",\"div\",null,{\"className\":\"flex flex-col md:flex-row gap-6 items-start\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex-shrink-0 w-16 h-16 rounded-2xl bg-primary text-primary-foreground flex items-center justify-center text-2xl font-bold\",\"children\":\"3\"}],[\"$\",\"div\",null,{\"className\":\"flex-1\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex items-start gap-3 mb-3\",\"children\":[[\"$\",\"svg\",null,{\"xmlns\":\"http://www.w3.org/2000/svg\",\"width\":24,\"height\":24,\"viewBox\":\"0 0 24 24\",\"fill\":\"none\",\"stroke\":\"currentColor\",\"strokeWidth\":2,\"strokeLinecap\":\"round\",\"strokeLinejoin\":\"round\",\"className\":\"lucide lucide-git-branch h-6 w-6 text-primary flex-shrink-0 mt-1\",\"children\":[[\"$\",\"line\",\"17qcm7\",{\"x1\":\"6\",\"x2\":\"6\",\"y1\":\"3\",\"y2\":\"15\"}],[\"$\",\"circle\",\"1h7g24\",{\"cx\":\"18\",\"cy\":\"6\",\"r\":\"3\"}],[\"$\",\"circle\",\"fqmcym\",{\"cx\":\"6\",\"cy\":\"18\",\"r\":\"3\"}],[\"$\",\"path\",\"n2h4wq\",{\"d\":\"M18 9a9 9 0 0 1-9 9\"}],\"$undefined\"]}],[\"$\",\"h3\",null,{\"className\":\"text-2xl font-bold\",\"children\":\"Build a DAG (partial order) with dominance weights\"}]]}],[\"$\",\"p\",null,{\"className\":\"text-muted-foreground mb-4\",\"children\":\"Organize criteria into levels where higher-priority wins can't be gamed by stacking lower-level features. Assign level-dominant weights or use lexicographic vectors.\"}],[\"$\",\"div\",null,{\"className\":\"rounded-2xl border bg-card text-card-foreground shadow-lg\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex flex-col space-y-1.5 p-6\",\"children\":[[\"$\",\"h3\",null,{\"className\":\"font-semibold tracking-tight text-lg\",\"children\":\"How the DAG Creates Dominance\"}],[\"$\",\"p\",null,{\"className\":\"text-sm text-muted-foreground\",\"children\":\"The partial order determines which laptops are optimal\"}]]}],[\"$\",\"div\",null,{\"className\":\"p-6 pt-0 space-y-4 text-sm\",\"children\":[[\"$\",\"div\",null,{\"className\":\"p-4 bg-gradient-to-r from-blue-50 to-indigo-50 dark:from-blue-950 dark:to-indigo-950 rounded-lg border-l-4 border-blue-500\",\"children\":[[\"$\",\"h4\",null,{\"className\":\"font-semibold text-foreground mb-2\",\"children\":\"Preference Comparisons\"}],[\"$\",\"p\",null,{\"className\":\"text-muted-foreground mb-3\",\"children\":\"The system evaluates each laptop against the partial order preferences. A laptop satisfying more preference comparisons dominates one satisfying fewer.\"}],[\"$\",\"div\",null,{\"className\":\"space-y-2 text-xs\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex items-start gap-2\",\"children\":[[\"$\",\"span\",null,{\"className\":\"text-green-600 dark:text-green-400\",\"children\":\"â\"}],[\"$\",\"span\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Gray\"}],\" is preferred over Black (meets 'gray \u003e black')\"]}]]}],[\"$\",\"div\",null,{\"className\":\"flex items-start gap-2\",\"children\":[[\"$\",\"span\",null,{\"className\":\"text-green-600 dark:text-green-400\",\"children\":\"â\"}],[\"$\",\"span\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Intel\"}],\" is preferred over AMD (meets 'intel \u003e amd')\"]}]]}],[\"$\",\"div\",null,{\"className\":\"flex items-start gap-2\",\"children\":[[\"$\",\"span\",null,{\"className\":\"text-green-600 dark:text-green-400\",\"children\":\"â\"}],[\"$\",\"span\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"New\"}],\" is preferred over Refurbished (meets 'new \u003e refurbished')\"]}]]}],[\"$\",\"div\",null,{\"className\":\"flex items-start gap-2\",\"children\":[[\"$\",\"span\",null,{\"className\":\"text-purple-600 dar
1k:text-purple-400\",\"children\":\"â\"}],[\"$\",\"span\",null,{\"className\":\"text-purple-700 dark:text-purple-300\",\"children\":\"Implicit preferences (rating, battery, etc.) only used to break ties\"}]]}]]}]]}],[\"$\",\"div\",null,{\"className\":\"p-4 bg-green-50 dark:bg-green-950 rounded-lg border-l-4 border-green-500\",\"children\":[[\"$\",\"h4\",null,{\"className\":\"font-semibold text-foreground mb-2\",\"children\":\"â Gaming-Resistant Property\"}],[\"$\",\"p\",null,{\"className\":\"text-muted-foreground\",\"children\":[\"A laptop matching more DAG preferences (Gray + Intel + New = 3/3 preferences met) will \",[\"$\",\"strong\",null,{\"children\":\"always dominate\"}],\"one matching fewer preferences (Black + Intel + New = 2/3 preferences met), even if the latter excels on implicit factors like price or battery life.\",[\"$\",\"span\",null,{\"className\":\"block mt-2 text-xs italic\",\"children\":\"The partial order ensures explicit preferences cannot be overridden by implicit tie-breakers.\"}]]}]]}],[\"$\",\"div\",null,{\"className\":\"p-4 bg-amber-50 dark:bg-amber-950 rounded-lg border-l-4 border-amber-500\",\"children\":[[\"$\",\"h4\",null,{\"className\":\"font-semibold text-foreground mb-2\",\"children\":\"How It Differs From Traditional Ranking\"}],[\"$\",\"p\",null,{\"className\":\"text-muted-foreground text-xs\",\"children\":[\"Traditional ML ranking might weight features equally: Price (30%), Battery (25%), Rating (20%), Color (15%), Processor (10%). A cheap laptop with great battery/rating could beat your preferred Gray+Intel choice. With dominance-based DAG, explicit preferences (color, processor, condition) \",[\"$\",\"strong\",null,{\"children\":\"always\"}],\" dominate implicit tie-breakers.\"]}]]}]]}]]}]]}]]}],[\"$\",\"div\",null,{\"className\":\"flex flex-col md:flex-row gap-6 items-start\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex-shrink-0 w-16 h-16 rounded-2xl bg-primary text-primary-foreground flex items-center justify-center text-2xl font-bold\",\"children\":\"4\"}],[\"$\",\"div\",null,{\"className\":\"flex-1\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex items-start gap-3 mb-3\",\"children\":[[\"$\",\"svg\",null,{\"xmlns\":\"http://www.w3.org/2000/svg\",\"width\":24,\"height\":24,\"viewBox\":\"0 0 24 24\",\"fill\":\"none\",\"stroke\":\"currentColor\",\"strokeWidth\":2,\"strokeLinecap\":\"round\",\"strokeLinejoin\":\"round\",\"className\":\"lucide lucide-target h-6 w-6 text-primary flex-shrink-0 mt-1\",\"children\":[[\"$\",\"circle\",\"1mglay\",{\"cx\":\"12\",\"cy\":\"12\",\"r\":\"10\"}],[\"$\",\"circle\",\"1vlfrh\",{\"cx\":\"12\",\"cy\":\"12\",\"r\":\"6\"}],[\"$\",\"circle\",\"1c9p78\",{\"cx\":\"12\",\"cy\":\"12\",\"r\":\"2\"}],\"$undefined\"]}],[\"$\",\"h3\",null,{\"className\":\"text-2xl font-bold\",\"children\":\"Score survivors; return optimal or co-optimal results\"}]]}],[\"$\",\"p\",null,{\"className\":\"text-muted-foreground mb-4\",\"children\":\"Apply the preference DAG to rank surviving candidates. Return the single optimal choice, or all co-optimal results (if multiple items tie at the top) with a level-by-level explanation.\"}],[\"$\",\"div\",null,{\"className\":\"space-y-4\",\"children\":[[\"$\",\"div\",null,{\"className\":\"rounded-2xl bg-card text-card-foreground shadow-lg border-2 border-green-500\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex flex-col space-y-1.5 p-6 bg-green-50 dark:bg-green-950\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex items-center gap-2\",\"children\":[[\"$\",\"span\",null,{\"className\":\"text-2xl\",\"children\":\"ð\"}],[\"$\",\"h3\",null,{\"className\":\"font-semibold tracking-tight text-lg\",\"children\":\"Optimal Result\"}]]}],[\"$\",\"p\",null,{\"className\":\"text-sm text-muted-foreground\",\"children\":\"Dell XPS 13 Plus - Gray, Intel i7-1360P, New\"}]]}],[\"$\",\"div\",null,{\"className\":\"p-6 space-y-3 pt-4\",\"children\":[[\"$\",\"div\",null,{\"className\":\"grid grid-cols-2 gap-3 text-sm mb-4\",\"children\":[[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Price:\"}],\" $949\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"RAM:\"}],\" 16GB\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Storage:\"}],\" 512GB SSD\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Weight:\"}],\" 1.24kg\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"USB-C:\"}],\" 2 ports\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Wi-Fi:\"}],\" 6E\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Battery:\"}],\" 12h\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Rating:\"}],\" 4.7/5\"]}]]}],[\"$\",\"div\",null,{\"className\":\"p-3 border-l-4 border-blue-500 bg-blue-50 dark:bg-blue-950 rounded text-sm\",\"children\":[[\"$\",\"p\",null,{\"className\":\"font-semibold text-foreground mb-1\",\"children\":\"â Mandatory constraints:\"}],[\"$\",\"p\",null,{\"className\":\"text-muted-foreground\",\"children\":\"Passed all: priceâ¤$1000, RAMâ¥16GB, storageâ¥512GB, weightâ¤1.5kg, 2ÃUSB-C, Wi-Fi 6E, delivers by Nov 1\"}]]}],[\"$\",\"div\",null,{\"className\":\"p-3 border-l-4 border-primary bg-primary/5 rounded text-sm\",\"children\":[[\"$\",\"p\",null,{\"className\":\"font-semibold text-foreground mb-1\",\"children\":\"â DAG Preferences:\"}],[\"$\",\"div\",null,{\"className\":\"text-muted-foreground space-y-1 mt-2\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex items-center gap-2\",\"children\":[[\"$\",\"span\",null,{\"className\":\"text-green-600 dark:text-green-400\",\"children\":\"â\"}],[\"$\",\"span\",null,{\"children\":\"Gray color (meets 'gray \u003e black')\"}]]}],[\"$\",\"div\",null,{\"className\":\"flex items-center gap-2\",\"children\":[[\"$\",\"span\",null,{\"className\":\"text-green-600 dark:text-green-400\",\"children\":\"â\"}],[\"$\",\"span\",null,{\"children\":\"Intel i7-1360P (meets 'intel \u003e amd')\"}]]}],[\"$\",\"div\",null,{\"className\":\"flex items-center gap-2\",\"children\":[[\"$\",\"span\",null,{\"className\":\"text-green-600 dark:text-green-400\",\"children\":\"â\"}],[\"$\",\"span\",null,{\"children\":\"New condition (meets 'new \u003e refurbished')\"}]]}]]}]]}],[\"$\",\"div\",null,{\"className\":\"p-3 border-l-4 border-green-500 bg-green-50 dark:bg-green-950 rounded text-sm\",\"children\":[[\"$\",\"p\",null,{\"className\":\"font-semibold text-green-900 dark:text-green-100 mb-1\",\"children\":\"ð Why optimal:\"}],[\"$\",\"p\",null,{\"className\":\"text-green-800 dark:text-green-200\",\"children\":\"This laptop satisfies all mandatory constraints AND all three DAG preference comparisons (3/3 met). No other candidate dominates it on the partial order. It represents the unique optimal choice.\"}]]}]]}]]}],[\"$\",\"div\",null,{\"className\":\"rounded-2xl bg-card text-card-foreground shadow-lg border-2 border-orange-300\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex flex-col space-y-1.5 p-6 bg-orange-50 dark:bg-orange-950\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex items-center gap-2\",\"children\":[[\"$\",\"span\",null,{\"className\":\"text-2xl\",\"children\":\"â ï¸\"}],[\"$\",\"h3\",null,{\"className\":\"font-semibold tracking-tight text-lg\",\"children\":\"Non-Optimal Result #1\"}]]}],[\"$\",\"p\",null,{\"className\":\"text-sm text-muted-foreground\",\"children\":\"HP Spectre x360 - Black, Intel i7-1355U, New\"}]]}],[\"$\",\"div\",null,{\"className\":\"p-6 space-y-3 pt-4\",\"children\":[[\"$\",\"div\",null,{\"className\":\"grid grid-cols-2 gap-3 text-sm mb-4\",\"children\":[[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Price
1:\"}],\" $929\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"RAM:\"}],\" 16GB\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Storage:\"}],\" 512GB SSD\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Weight:\"}],\" 1.36kg\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"USB-C:\"}],\" 2 ports\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Wi-Fi:\"}],\" 6E\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Battery:\"}],\" 14h\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Rating:\"}],\" 4.8/5\"]}]]}],[\"$\",\"div\",null,{\"className\":\"p-3 border-l-4 border-blue-500 bg-blue-50 dark:bg-blue-950 rounded text-sm\",\"children\":[[\"$\",\"p\",null,{\"className\":\"font-semibold text-foreground mb-1\",\"children\":\"â Mandatory constraints:\"}],[\"$\",\"p\",null,{\"className\":\"text-muted-foreground\",\"children\":\"Passed all constraints\"}]]}],[\"$\",\"div\",null,{\"className\":\"p-3 border-l-4 border-primary bg-primary/5 rounded text-sm\",\"children\":[[\"$\",\"p\",null,{\"className\":\"font-semibold text-foreground mb-1\",\"children\":\"DAG Preferences:\"}],[\"$\",\"div\",null,{\"className\":\"text-muted-foreground space-y-1 mt-2\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex items-center gap-2\",\"children\":[[\"$\",\"span\",null,{\"className\":\"text-red-600 dark:text-red-400\",\"children\":\"â\"}],[\"$\",\"span\",null,{\"children\":\"Black color (fails 'gray \u003e black')\"}]]}],[\"$\",\"div\",null,{\"className\":\"flex items-center gap-2\",\"children\":[[\"$\",\"span\",null,{\"className\":\"text-green-600 dark:text-green-400\",\"children\":\"â\"}],[\"$\",\"span\",null,{\"children\":\"Intel i7-1355U (meets 'intel \u003e amd')\"}]]}],[\"$\",\"div\",null,{\"className\":\"flex items-center gap-2\",\"children\":[[\"$\",\"span\",null,{\"className\":\"text-green-600 dark:text-green-400\",\"children\":\"â\"}],[\"$\",\"span\",null,{\"children\":\"New condition (meets 'new \u003e refurbished')\"}]]}]]}]]}],[\"$\",\"div\",null,{\"className\":\"p-3 border-l-4 border-orange-500 bg-orange-50 dark:bg-orange-950 rounded text-sm\",\"children\":[[\"$\",\"p\",null,{\"className\":\"font-semibold text-orange-900 dark:text-orange-100 mb-1\",\"children\":\"Why NOT optimal:\"}],[\"$\",\"p\",null,{\"className\":\"text-orange-800 dark:text-orange-200\",\"children\":[\"Even though this laptop is cheaper ($929 vs $949) and has better battery life (14h vs 12h), it's \",[\"$\",\"strong\",null,{\"children\":\"dominated\"}],\" by the optimal choice because it meets only 2/3 DAG preferences (black instead of gray). Implicit advantages (price, battery) cannot overcome explicit preference differences in the partial order.\"]}]]}]]}]]}],[\"$\",\"div\",null,{\"className\":\"rounded-2xl bg-card text-card-foreground shadow-lg border-2 border-red-300\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex flex-col space-y-1.5 p-6 bg-red-50 dark:bg-red-950\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex items-center gap-2\",\"children\":[[\"$\",\"span\",null,{\"className\":\"text-2xl\",\"children\":\"â\"}],[\"$\",\"h3\",null,{\"className\":\"font-semibold tracking-tight text-lg\",\"children\":\"Non-Optimal Result #2\"}]]}],[\"$\",\"p\",null,{\"className\":\"text-sm text-muted-foreground\",\"children\":\"Lenovo ThinkPad X1 - Gray, AMD Ryzen 7, New\"}]]}],[\"$\",\"div\",null,{\"className\":\"p-6 space-y-3 pt-4\",\"children\":[[\"$\",\"div\",null,{\"className\":\"grid grid-cols-2 gap-3 text-sm mb-4\",\"children\":[[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Price:\"}],\" $899\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"RAM:\"}],\" 16GB\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Storage:\"}],\" 512GB SSD\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Weight:\"}],\" 1.19kg\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"USB-C:\"}],\" 2 ports\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Wi-Fi:\"}],\" 6E\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Battery:\"}],\" 15h\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Rating:\"}],\" 4.9/5\"]}]]}],[\"$\",\"div\",null,{\"className\":\"p-3 border-l-4 border-blue-500 bg-blue-50 dark:bg-blue-950 rounded text-sm\",\"children\":[[\"$\",\"p\",null,{\"className\":\"font-semibold text-foreground mb-1\",\"children\":\"â Mandatory constraints:\"}],[\"$\",\"p\",null,{\"className\":\"text-muted-foreground\",\"children\":\"Passed all constraints\"}]]}],[\"$\",\"div\",null,{\"className\":\"p-3 border-l-4 border-primary bg-primary/5 rounded text-sm\",\"children\":[[\"$\",\"p\",null,{\"className\":\"font-semibold text-foreground mb-1\",\"children\":\"DAG Preferences:\"}],[\"$\",\"div\",null,{\"className\":\"text-muted-foreground space-y-1 mt-2\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex items-center gap-2\",\"children\":[[\"$\",\"span\",null,{\"className\":\"text-green-600 dark:text-green-400\",\"children\":\"â\"}],[\"$\",\"span\",null,{\"children\":\"Gray color (meets 'gray \u003e black')\"}]]}],[\"$\",\"div\",null,{\"className\":\"flex items-center gap-2\",\"children\":[[\"$\",\"span\",null,{\"className\":\"text-red-600 dark:text-red-400\",\"children\":\"â\"}],[\"$\",\"span\",null,{\"children\":\"AMD Ryzen 7 (fails 'intel \u003e amd')\"}]]}],[\"$\",\"div\",null,{\"className\":\"flex items-center gap-2\",\"children\":[[\"$\",\"span\",null,{\"className\":\"text-green-600 dark:text-green-400\",\"children\":\"â\"}],[\"$\",\"span\",null,{\"children\":\"New condition (meets 'new \u003e refurbished')\"}]]}]]}]]}],[\"$\",\"div\",null,{\"className\":\"p-3 border-l-4 border-red-500 bg-red-50 dark:bg-red-950 rounded text-sm\",\"children\":[[\"$\",\"p\",null,{\"className\":\"font-semibold text-red-900 dark:text-red-100 mb-1\",\"children\":\"Why NOT optimal:\"}],[\"$\",\"p\",null,{\"className\":\"text-red-800 dark:text-red-200\",\"children\":[\"Despite being cheaper ($899), lighter (1.19kg), having better battery (15h), and higher rating (4.9/5), this laptop is \",[\"$\",\"strong\",null,{\"children\":\"dominated\"}],\" by the optimal choice because it meets only 2/3 DAG preferences (AMD instead of Intel). Gaming-resistant: you can't \\\"win\\\" by stacking implicit features (price, weight, battery) if you fail explicit DAG preferences.\"]}]]}]]}]]}],[\"$\",\"div\",null,{\"className\":\"rounded-2xl bg-card text-card-foreground shadow-lg border-2 border-gray-400\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex flex-col space-y-1.5 p-6 bg-gray-50 dark:bg-gray-900\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex items-center gap-2\",\"children\":[[\"$\",\"span\",null,{\"className\":\"text-2xl\",\"children\":\"ð«\"}],[\"$\",\"h3\",null,{\"className\":\"font-semibold tracking-tight text-lg\",\"children\":\"Excluded Result\"}]]}],[\"$\",\"p\",null,{\"className\":\"text-sm text-muted-foreground\",\"children\":\"ASUS ZenBook - Silver, Intel i7-1355U, New\"}]]}],[\"$\",\"div\",null,{\"className\":\"p-6 space-y-3 pt-4\",\"children\":[[\"$\",\"div\",null,{\"className\":\"grid grid-cols-2 gap-3 text-sm mb-4\",\"children\":[[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Price
1:\"}],\" $1,099\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"RAM:\"}],\" 16GB\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Storage:\"}],\" 512GB SSD\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Weight:\"}],\" 1.29kg\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"USB-C:\"}],\" 2 ports\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Wi-Fi:\"}],\" 6E\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Battery:\"}],\" 13h\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"strong\",null,{\"children\":\"Rating:\"}],\" 4.6/5\"]}]]}],[\"$\",\"div\",null,{\"className\":\"p-3 border-l-4 border-red-500 bg-red-50 dark:bg-red-950 rounded text-sm\",\"children\":[[\"$\",\"p\",null,{\"className\":\"font-semibold text-foreground mb-1\",\"children\":\"â Mandatory constraints:\"}],[\"$\",\"p\",null,{\"className\":\"text-muted-foreground\",\"children\":\"Failed: price=$1,099 \u003e $1,000 maximum\"}]]}],[\"$\",\"div\",null,{\"className\":\"p-3 border-l-4 border-gray-500 bg-gray-50 dark:bg-gray-900 rounded text-sm\",\"children\":[[\"$\",\"p\",null,{\"className\":\"font-semibold text-gray-900 dark:text-gray-100 mb-1\",\"children\":\"Why EXCLUDED:\"}],[\"$\",\"p\",null,{\"className\":\"text-gray-800 dark:text-gray-200\",\"children\":\"This laptop violates a hard constraint (price \u003e $1,000), so it's eliminated before preference evaluation. Hard constraints are non-negotiable filters applied first.\"}]]}]]}]]}]]}]]}]]}]]}]]}]}],[\"$\",\"section\",null,{\"className\":\"py-16 sm:py-24 bg-muted/30\",\"children\":[\"$\",\"div\",null,{\"className\":\"mx-auto max-w-7xl px-6 lg:px-8\",\"children\":[[\"$\",\"div\",null,{\"className\":\"mx-auto max-w-2xl text-center mb-12\",\"children\":[[\"$\",\"h2\",null,{\"className\":\"text-3xl font-bold tracking-tight sm:text-4xl mb-4\",\"children\":\"Why This Is Not a Trivial Problem\"}],[\"$\",\"p\",null,{\"className\":\"text-lg text-muted-foreground\",\"children\":\"Finding optimal solutions with preferences requires solving fundamental computational challenges\"}]]}],[\"$\",\"div\",null,{\"className\":\"mx-auto max-w-5xl space-y-8\",\"children\":[[\"$\",\"div\",null,{\"className\":\"rounded-2xl border bg-card text-card-foreground shadow-lg border-l-4 border-red-500\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex flex-col space-y-1.5 p-6\",\"children\":[[\"$\",\"h3\",null,{\"className\":\"font-semibold tracking-tight text-2xl\",\"children\":\"Traditional Approaches Require Expensive Dominance Testing\"}],[\"$\",\"p\",null,{\"className\":\"text-sm text-muted-foreground\",\"children\":\"Without our invention, systems must compare every solution against all others\"}]]}],[\"$\",\"div\",null,{\"className\":\"p-6 pt-0 space-y-4\",\"children\":[[\"$\",\"p\",null,{\"className\":\"text-muted-foreground\",\"children\":[\"When a system \\\"sees\\\" a potential solution, traditional approaches \",[\"$\",\"strong\",null,{\"children\":\"cannot\"}],\" determine if it's optimal without comparing it against \",[\"$\",\"strong\",null,{\"children\":\"all previously found solutions\"}],\" via computationally expensive dominance testing procedures.\"]}],[\"$\",\"div\",null,{\"className\":\"bg-red-50 dark:bg-red-950 p-4 rounded-lg border border-red-200 dark:border-red-800\",\"children\":[[\"$\",\"h4\",null,{\"className\":\"font-semibold text-red-900 dark:text-red-100 mb-2\",\"children\":\"The Dominance Testing Problem:\"}],[\"$\",\"ul\",null,{\"className\":\"space-y-2 text-sm text-red-800 dark:text-red-200\",\"children\":[[\"$\",\"li\",null,{\"children\":[\"⢠For \",[\"$\",\"strong\",null,{\"children\":\"each solution\"}],\", perform a dominance test with \",[\"$\",\"strong\",null,{\"children\":\"all other solutions\"}]]}],[\"$\",\"li\",null,{\"children\":[\"⢠Computational cost grows \",[\"$\",\"strong\",null,{\"children\":\"quadratically (O(N²))\"}],\" with the number of solutions\"]}],[\"$\",\"li\",null,{\"children\":\"⢠With 1,000 solutions: ~1,000,000 comparisons needed\"}],[\"$\",\"li\",null,{\"children\":\"⢠With 10,000 solutions: ~100,000,000 comparisons needed\"}]]}]]}],[\"$\",\"p\",null,{\"className\":\"text-muted-foreground\",\"children\":[[\"$\",\"strong\",null,{\"children\":\"Worse still:\"}],\" Such approaches require that \",[\"$\",\"strong\",null,{\"children\":\"all solutions are already available\"}],\" before starting to calculate optimal results. This is \",[\"$\",\"strong\",null,{\"children\":\"impractical and infeasible\"}],\" in real-world, industrial use cases dealing with:\"]}],[\"$\",\"div\",null,{\"className\":\"grid md:grid-cols-3 gap-4\",\"children\":[[\"$\",\"div\",null,{\"className\":\"p-3 bg-muted rounded-lg\",\"children\":[[\"$\",\"div\",null,{\"className\":\"font-semibold mb-1\",\"children\":\"Huge Volumes\"}],[\"$\",\"div\",null,{\"className\":\"text-xs text-muted-foreground\",\"children\":\"Millions of products from multiple sources\"}]]}],[\"$\",\"div\",null,{\"className\":\"p-3 bg-muted rounded-lg\",\"children\":[[\"$\",\"div\",null,{\"className\":\"font-semibold mb-1\",\"children\":\"Progressive Arrival\"}],[\"$\",\"div\",null,{\"className\":\"text-xs text-muted-foreground\",\"children\":\"Solutions appear incrementally over time\"}]]}],[\"$\",\"div\",null,{\"className\":\"p-3 bg-muted rounded-lg\",\"children\":[[\"$\",\"div\",null,{\"className\":\"font-semibold mb-1\",\"children\":\"Multiple Data Sources\"}],[\"$\",\"div\",null,{\"className\":\"text-xs text-muted-foreground\",\"children\":\"Parallel queries to different databases/APIs\"}]]}]]}]]}]]}],[\"$\",\"div\",null,{\"className\":\"rounded-2xl border bg-card text-card-foreground shadow-lg border-l-4 border-amber-500\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex flex-col space-y-1.5 p-6\",\"children\":[[\"$\",\"h3\",null,{\"className\":\"font-semibold tracking-tight text-2xl\",\"children\":\"Trivial Scoring Methods Are Mathematically Wrong\"}],[\"$\",\"p\",null,{\"className\":\"text-sm text-muted-foreground\",\"children\":\"Simple linear scoring leads to fal
1sely \\\"optimal\\\" solutions\"}]]}],[\"$\",\"div\",null,{\"className\":\"p-6 pt-0 space-y-4\",\"children\":[[\"$\",\"p\",null,{\"className\":\"text-muted-foreground\",\"children\":[\"A naive approach might try to compute weights for DAG nodes by simply summing them up following the partial order, or using linearly increasing weights from top to bottom preferences. \",[\"$\",\"strong\",null,{\"children\":\"This is mathematically incorrect\"}],\" and leads to falsely \\\"optimal\\\" solutions.\"]}],[\"$\",\"div\",null,{\"className\":\"bg-amber-50 dark:bg-amber-950 p-4 rounded-lg border border-amber-200 dark:border-amber-800\",\"children\":[[\"$\",\"h4\",null,{\"className\":\"font-semibold text-amber-900 dark:text-amber-100 mb-3\",\"children\":\"Example: Why Linear Weights Fail\"}],[\"$\",\"p\",null,{\"className\":\"text-sm text-amber-800 dark:text-amber-200 mb-3\",\"children\":\"Consider a DAG with preferences at different levels. A trivial system using simple additive or linear weights will produce incorrect results where lower-level preferences can overwhelm higher-level ones.\"}],[\"$\",\"div\",null,{\"className\":\"bg-red-100 dark:bg-red-900 p-3 rounded text-sm\",\"children\":[[\"$\",\"strong\",null,{\"className\":\"text-red-900 dark:text-red-100\",\"children\":\"The Problem:\"}],[\"$\",\"p\",null,{\"className\":\"text-red-800 dark:text-red-200 mt-1\",\"children\":[\"With linear weighting, a solution satisfying multiple lower-priority preferences can incorrectly outscore a solution satisfying a single higher-priority preference. \",[\"$\",\"strong\",null,{\"children\":\"This violates the dominance hierarchy and produces falsely \\\"optimal\\\" solutions.\"}]]}]]}]]}],[\"$\",\"p\",null,{\"className\":\"text-sm text-muted-foreground italic\",\"children\":\"This trivial approach only works correctly in the special case of a simple linear chain of preferences, but fails for general DAGs with multiple roots and parallel branches.\"}]]}]]}]]}]]}]}],[\"$\",\"section\",null,{\"className\":\"py-16 sm:py-24\",\"children\":[\"$\",\"div\",null,{\"className\":\"mx-auto max-w-7xl px-6 lg:px-8\",\"children\":[[\"$\",\"div\",null,{\"className\":\"mx-auto max-w-2xl text-center mb-12\",\"children\":[[\"$\",\"h2\",null,{\"className\":\"text-3xl font-bold tracking-tight sm:text-4xl mb-4\",\"children\":\"How Our Invention Solves These Problems\"}],[\"$\",\"p\",null,{\"className\":\"text-lg text-muted-foreground\",\"children\":\"Our patent-pending method enables progressive scoring without dominance testing\"}]]}],[\"$\",\"div\",null,{\"className\":\"mx-auto max-w-5xl space-y-8\",\"children\":[\"$\",\"div\",null,{\"className\":\"rounded-2xl text-card-foreground shadow-lg border-2 border-green-500 bg-green-50/30 dark:bg-green-950/20\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex flex-col space-y-1.5 p-6\",\"children\":[[\"$\",\"h3\",null,{\"className\":\"font-semibold tracking-tight text-2xl\",\"children\":\"Key Innovation: Direct Optimality Scoring\"}],[\"$\",\"p\",null,{\"className\":\"text-sm text-muted-foreground\",\"children\":\"Score each solution independently using only the DAG structure\"}]]}],[\"$\",\"div\",null,{\"className\":\"p-6 pt-0 space-y-6\",\"children\":[[\"$\",\"p\",null,{\"className\":\"text-muted-foreground\",\"children\":[\"When the system \\\"sees\\\" a potential solution, it can produce an \",[\"$\",\"strong\",null,{\"children\":\"optimality score calculated using only the current solution and the DAG\"}],\" â no comparison with other solutions needed!\"]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"h4\",null,{\"className\":\"font-bold text-lg mb-3\",\"children\":\"â
Properties Guaranteed by Our System:\"}],[\"$\",\"div\",null,{\"className\":\"grid md:grid-cols-2 gap-4\",\"children\":[[\"$\",\"div\",null,{\"className\":\"bg-blue-50 dark:bg-blue-950 p-4 rounded-lg border border-blue-200 dark:border-blue-800\",\"children\":[[\"$\",\"div\",null,{\"className\":\"font-semibold text-blue-900 dark:text-blue-100 mb-2\",\"children\":\"ð Pareto-Optimality\"}],[\"$\",\"p\",null,{\"className\":\"text-sm text-blue-800 dark:text-blue-200\",\"children\":\"Solutions with maximum score are guaranteed to be Pareto-optimal with respect to the DAG preferences. No other solution dominates them.\"}]]}],[\"$\",\"div\",null,{\"className\":\"bg-purple-50 dark:bg-purple-950 p-4 rounded-lg border border-purple-200 dark:border-purple-800\",\"children\":[[\"$\",\"div\",null,{\"className\":\"font-semibold text-purple-900 dark:text-purple-100 mb-2\",\"children\":\"âï¸ Equivalence Property\"}],[\"$\",\"p\",null,{\"className\":\"text-sm text-purple-800 dark:text-purple-200\",\"children\":\"Solutions with equal scores are truly equivalent with respect to preferences â they satisfy the same preference comparisons.\"}]]}],[\"$\",\"div\",null,{\"className\":\"bg-green-50 dark:bg-green-950 p-4 rounded-lg border border-green-200 dark:border-green-800\",\"children\":[[\"$\",\"div\",null,{\"className\":\"font-semibold text-green-900 dark:text-green-100 mb-2\",\"children\":\"â¡ Progressive Computation\"}],[\"$\",\"p\",null,{\"className\":\"text-sm text-green-800 dark:text-green-200\",\"children\":\"Solutions can arrive progressively (from multiple data sources, streaming APIs, etc.) and be scored immediately without waiting for all solutions.\"}]]}
1],[\"$\",\"div\",null,{\"className\":\"bg-orange-50 dark:bg-orange-950 p-4 rounded-lg border border-orange-200 dark:border-orange-800\",\"children\":[[\"$\",\"div\",null,{\"className\":\"font-semibold text-orange-900 dark:text-orange-100 mb-2\",\"children\":\"â»ï¸ No Recalculation Needed\"}],[\"$\",\"p\",null,{\"className\":\"text-sm text-orange-800 dark:text-orange-200\",\"children\":\"When new solutions arrive, their scores remain valid. No need to recalculate or restart the computation from scratch.\"}]]}]]}]]}],[\"$\",\"div\",null,{\"className\":\"bg-gradient-to-r from-cyan-50 to-blue-50 dark:from-cyan-950 dark:to-blue-950 p-5 rounded-lg border-2 border-cyan-500\",\"children\":[[\"$\",\"h4\",null,{\"className\":\"font-bold text-lg mb-3 text-cyan-900 dark:text-cyan-100\",\"children\":\"ð Dramatic Computational Improvement\"}],[\"$\",\"div\",null,{\"className\":\"grid md:grid-cols-2 gap-4 mb-4\",\"children\":[[\"$\",\"div\",null,{\"className\":\"bg-red-100 dark:bg-red-900 p-4 rounded border-2 border-red-400\",\"children\":[[\"$\",\"div\",null,{\"className\":\"font-semibold text-red-900 dark:text-red-100 mb-2\",\"children\":\"â Traditional Approach\"}],[\"$\",\"div\",null,{\"className\":\"font-mono text-xl text-red-700 dark:text-red-300 mb-2\",\"children\":\"O(N² à V)\"}],[\"$\",\"p\",null,{\"className\":\"text-xs text-red-800 dark:text-red-200\",\"children\":\"Quadratic growth: must compare each solution against all others (N²) for each preference criterion (V)\"}]]}],[\"$\",\"div\",null,{\"className\":\"bg-green-100 dark:bg-green-900 p-4 rounded border-2 border-green-400\",\"children\":[[\"$\",\"div\",null,{\"className\":\"font-semibold text-green-900 dark:text-green-100 mb-2\",\"children\":\"â
Our System\"}],[\"$\",\"div\",null,{\"className\":\"font-mono text-xl text-green-700 dark:text-green-300 mb-2\",\"children\":\"O(N à V + N log N)\"}],[\"$\",\"p\",null,{\"className\":\"text-xs text-green-800 dark:text-green-200\",\"children\":\"Near-linear growth: score each solution once (N à V), then sort results (N log N)\"}]]}]]}],[\"$\",\"div\",null,{\"className\":\"bg-white dark:bg-gray-900 p-4 rounded\",\"children\":[[\"$\",\"h5\",null,{\"className\":\"font-semibold mb-2 text-sm\",\"children\":\"Real-World Impact:\"}],[\"$\",\"div\",null,{\"className\":\"space-y-2 text-xs text-muted-foreground\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex justify-between items-center\",\"children\":[[\"$\",\"span\",null,{\"children\":\"1,000 solutions, 10 preferences:\"}],[\"$\",\"span\",null,{\"className\":\"font-semibold\",\"children\":[[\"$\",\"span\",null,{\"className\":\"text-red-600\",\"children\":\"10M ops\"}],\" â \",[\"$\",\"span\",null,{\"className\":\"text-green-600\",\"children\":\"~10K ops\"}],[\"$\",\"span\",null,{\"className\":\"text-green-600 ml-2\",\"children\":\"(1,000à faster)\"}]]}]]}],[\"$\",\"div\",null,{\"className\":\"flex justify-between items-center\",\"children\":[[\"$\",\"span\",null,{\"children\":\"10,000 solutions, 20 preferences:\"}],[\"$\",\"span\",null,{\"className\":\"font-semibold\",\"children\":[[\"$\",\"span\",null,{\"className\":\"text-red-600\",\"children\":\"2B ops\"}],\" â \",[\"$\",\"span\",null,{\"className\":\"text-green-600\",\"children\":\"~200K ops\"}],[\"$\",\"span\",null,{\"className\":\"text-green-600 ml-2\",\"children\":\"(10,000à faster)\"}]]}]]}],[\"$\",\"div\",null,{\"className\":\"flex justify-between items-center\",\"children\":[[\"$\",\"span\",null,{\"children\":\"100,000 solutions, 50 preferences:\"}],[\"$\",\"span\",null,{\"className\":\"font-semibold\",\"children\":[[\"$\",\"span\",null,{\"className\":\"text-red-600\",\"children\":\"500B ops\"}],\" â \",[\"$\",\"span\",null,{\"className\":\"text-green-600\",\"children\":\"~5.8M ops\"}],[\"$\",\"span\",null,{\"className\":\"text-green-600 ml-2\",\"children\":\"(86,000à faster)\"}]]}]]}]]}]]}]]}],[\"$\",\"div\",null,{\"className\":\"bg-indigo-50 dark:bg-indigo-950 p-4 rounded-lg border border-indigo-200 dark:border-indigo-800\",\"children\":[[\"$\",\"h4\",null,{\"className\":\"font-bold text-lg mb-3 text-indigo-900 dark:text-indigo-100\",\"children\":\"âï¸ Parallel \u0026 Distributed Processing\"}],[\"$\",\"p\",null,{\"className\":\"text-sm text-indigo-800 dark:text-indigo-200 mb-3\",\"children\":\"Because each solution's optimality score is independent and stable, our system supports:\"}],[\"$\",\"ul\",null,{\"className\":\"space-y-2 text-sm text-indigo-800 dark:text-indigo-200\",\"children\":[[\"$\",\"li\",null,{\"children\":[\"⢠\",[\"$\",\"strong\",null,{\"children\":\"Parallel scoring\"}],\" of solutions across multiple CPU cores\"]}],[\"$\",\"li\",null,{\"children\":[\"⢠\",[\"$\",\"strong\",null,{\"children\":\"Distributed computation\"}],\" using \\\"partial rankers\\\" for solution subsets\"]}],[\"$\",\"li\",null,{\"children\":[\"⢠\",[\"$\",\"strong\",null,{\"children\":\"Incremental updates\"}],\" as new solutions arrive from different data sources\"]}],[\"$\",\"li\",null,{\"children\":[\"⢠\",[\"$\",\"strong\",null,{\"children\":\"Real-time ranking\"}],\" without reprocessing the entire solution set\"]}]]}]]}],[\"$\",\"div\",null,{\"className\":\"bg-gradient-to-r from-green-500 to-emerald-500 text-white p-6 rounded-lg text-center\",\"children\":[[\"$\",\"div\",null,{\"className\":\"text-4xl font-bold mb-2\",\"children\":\"Millisecond Response Times\"}],[\"$\",\"p\",null,{\"className\":\"text-green-100\",\"children\":\"Find optimal solutions from thousands of candidates in real-time, enabling live search, auto-suggest, and interactive user experiences\"}]]}]]}]]}]}]]}]}],[\"$\",\"section\",null,{\"className\":\"py-16 sm:py-24 bg-muted/30\",\"children\":[\"$\",\"div\",null,{\"className\":\"mx-auto max-w-7xl px-6 lg:px-8\",\"children\":[[\"$\",\"div\",null,{\"className\":\"mx-auto max-w-2xl text-center mb-12\",\"children\":[\"$\",\"h2\",null,{\"className\":\"text-3xl font-bold tracking-tight sm:text-4xl\",\"children\":\"Key properties\"}]}],[\"$\",\"div\",null,{\"className\":\"mx-auto max-w-5xl grid grid-cols-1 md:grid-cols-2 gap-6\",\"children\":[[\"$\",\"div\",null,{\"className\":\"rounded-2xl border bg-card text-card-foreground shadow-lg\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex flex-col space-y-1.5 p-6\",\"children\":[\"$\",\"h3\",null,{\"className\":\"text-2xl font-semibold leading-none tracking-tight\",\"children\":\"Provably optimal\"}]}],[\"$\",\"div\",null,{\"className\":\"p-6 pt-0\",\"children\":[\"$\",\"p\",null,{\"className\":\"text-sm text-muted-foreground\",\"children\":\"Dominance weights ensure higher-level wins can't be overturned by gaming lower levels. Patent-backed methodology (US 2025/0298806 A1).\"}]}]]}],[\"$\",\"div\",null,{\"className\":\"rounded-2xl border bg-card text-card-foreground shadow-lg\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex flex-col space-y-1.5 p-6\",\"children\":[\"$\",\"h3\",null,{\"className\":\"text-2xl font-semibold leading-none tracking-tight\",\"children\":\"Millisecond performance\"}]}],[\"$\",\"div\",null,{\"className\":\"p-6 pt-0\",\"children\":[\"$\",\"p\",null,{\"className\":\"text-sm text-muted-foreground\",\"children\":\"Constraint filters and efficie
1nt scoring return results in real-time. Fast enough for live search, auto-suggest, and A/B testing.\"}]}]]}],[\"$\",\"div\",null,{\"className\":\"rounded-2xl border bg-card text-card-foreground shadow-lg\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex flex-col space-y-1.5 p-6\",\"children\":[\"$\",\"h3\",null,{\"className\":\"text-2xl font-semibold leading-none tracking-tight\",\"children\":\"Explainable\"}]}],[\"$\",\"div\",null,{\"className\":\"p-6 pt-0\",\"children\":[\"$\",\"p\",null,{\"className\":\"text-sm text-muted-foreground\",\"children\":\"Every result includes a level-by-level rationale. Users and auditors can see exactly why each option won or lost.\"}]}]]}],[\"$\",\"div\",null,{\"className\":\"rounded-2xl border bg-card text-card-foreground shadow-lg\",\"children\":[[\"$\",\"div\",null,{\"className\":\"flex flex-col space-y-1.5 p-6\",\"children\":[\"$\",\"h3\",null,{\"className\":\"text-2xl font-semibold leading-none tracking-tight\",\"children\":\"Gaming-resistant\"}]}],[\"$\",\"div\",null,{\"className\":\"p-6 pt-0\",\"children\":[\"$\",\"p\",null,{\"className\":\"text-sm text-muted-foreground\",\"children\":\"Sellers can't boost rank by adding irrelevant features. Only improvements at the right dominance level matter.\"}]}]]}]]}]]}]}],[\"$\",\"section\",null,{\"className\":\"py-16 sm:py-24\",\"children\":[\"$\",\"div\",null,{\"className\":\"mx-auto max-w-7xl px-6 lg:px-8\",\"children\":[[\"$\",\"div\",null,{\"className\":\"text-center mb-10\",\"children\":[[\"$\",\"h2\",null,{\"className\":\"text-3xl font-bold mb-4\",\"children\":\"Comparison with State-of-the-Art Methods\"}],[\"$\",\"p\",null,{\"className\":\"text-lg text-muted-foreground max-w-3xl mx-auto\",\"children\":\"See how our Preference-DAG Optimality Scoring compares to existing approaches across key dimensions.\"}]]}],[\"$\",\"div\",null,{\"className\":\"overflow-x-auto mb-8\",\"children\":[\"$\",\"table\",null,{\"className\":\"w-full border-collapse text-sm\",\"children\":[[\"$\",\"thead\",null,{\"children\":[\"$\",\"tr\",null,{\"className\":\"bg-muted\",\"children\":[[\"$\",\"th\",null,{\"className\":\"border border-border p-3 text-left font-semibold min-w-[180px]\",\"children\":\"Approach\"}],[\"$\",\"th\",null,{\"className\":\"border border-border p-3 text-left font-semibold min-w-[120px]\",\"children\":\"Fast at Web Scale?\"}],[\"$\",\"th\",null,{\"className\":\"border border-border p-3 text-left font-semibold min-w-[120px]\",\"children\":\"Gaming-Resistant?\"}],[\"$\",\"th\",null,{\"className\":\"border border-border p-3 text-left font-semibold min-w-[120px]\",\"children\":\"Explainable?\"}],[\"$\",\"th\",null,{\"className\":\"border border-border p-3 text-left font-semibold min-w-[140px]\",\"children\":\"Provably Optimal?\"}],[\"$\",\"th\",null,{\"className\":\"border border-border p-3 text-left font-semibold min-w-[160px]\",\"children\":[\"Streaming-Compatible?\",[\"$\",\"sup\",null,{\"className\":\"text-[10px]\",\"children\":\"1\"}]]}],[\"$\",\"th\",null,{\"className\":\"border border-border p-3 text-left font-semibold min-w-[200px]\",\"children\":[\"Algorithmic Complexity\",[\"$\",\"sup\",null,{\"className\":\"text-[10px]\",\"children\":\"2\"}]]}]]}]}],[\"$\",\"tbody\",null,{\"children\":[[\"$\",\"tr\",null,{\"children\":[[\"$\",\"td\",null,{\"className\":\"border border-border p-3 font-medium\",\"children\":[\"Traditional ML Ranking\",[\"$\",\"br\",null,{}],[\"$\",\"span\",null,{\"className\":\"text-xs text-muted-foreground\",\"children\":\"Google, Amazon, Bing\"}]]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-green-600\",\"children\":\"â
Yes\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-amber-600\",\"children\":\"â ï¸ SEO arms race\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-amber-600\",\"children\":\"â ï¸ Opaque\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-red-600\",\"children\":\"â No proofs\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-amber-600\",\"children\":\"â ï¸ System-level only\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3 text-xs\",\"children\":[\"O(N log N)\",[\"$\",\"br\",null,{}],[\"$\",\"span\",null,{\"className\":\"text-muted-foreground\",\"children\":\"Fixed candidate set\"}]]}]]}],[\"$\",\"tr\",null,{\"className\":\"bg-muted/50\",\"children\":[[\"$\",\"td\",null,{\"className\":\"border border-border p-3 font-medium\",\"children\":[\"Recommendation Engines\",[\"$\",\"br\",null,{}],[\"$\",\"span\",null,{\"className\":\"text-xs text-muted-foreground\",\"children\":\"Netflix, Spotify, Amazon Recs\"}]]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-green-600\",\"children\":\"â
Yes\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-amber-600\",\"children\":\"â ï¸ Fraud detection\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-amber-600\",\"children\":\"â ï¸ Black-box\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-red-600\",\"children\":\"â Engagement proxy\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-amber-600\",\"children\":\"â ï¸ Session-level only\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3 text-xs\",\"children\":[\"O(N)\",[\"$\",\"br\",null,{}],[\"$\",\"span\",null,{\"className\":\"text-muted-foreground\",\"children\":\"Retrieval, not optimality\"}]]}]]}],[\"$\",\"tr\",null,{\"children\":[[\"$\",\"td\",null,{\"className\":\"border border-border p-3 font-medium\",\"children\":[\"MCDA / Pareto / Skyline\",[\"$\",\"br\",null,{}],[\"$\",\"span\",null,{\"className\":\"text-xs text-muted-foreground\",\"children\":\"AHP, TOPSIS, ELECTRE\"}]]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-amber-600\",\"children\":\"â ï¸ Limited scale\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-amber-600\",\"children\":\"â ï¸ Gameable\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-green-600\",\"children\":\"â
Yes\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-green-600\",\"children\":\"â
Pareto-optimal\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-red-600\",\"children\":\"â No\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3 text-xs\",\"children\":[\"O(N²ÃV)\",[\"$\",\"br\",null,{}],[\"$\",\"span\",null,{\"className\":\"text-muted-foreground\",\"children\":\"Dominance-testing based\"}]]}]]}],[\"$\",\"tr\",null,{\"className\":\"bg-muted/50\",\"children\":[[\"$\",\"td\",null,{\"className\":\"border border-border p-3 font-medium\",\"children\":[\"LLM-Enhanced Search\",[\"$\",\"br\",null,{}],[\"$\",\"span\",null,{\"className\":\"text-xs text-muted-foreground\",\"children\":\"Perplexity, ChatGPT+Search\"}]]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-amber-600\",\"children\":\"â ï¸ Slower (seconds)\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-amber-600\",\"children\":\"â ï¸ Prompt injection\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-amber-600\",\"children\":\"â ï¸ Citations only\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-red-600\",\"children\":\"â No guarantees\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-amber-600\",\"children\":\"â ï¸ Conversational only\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3 text-xs\",\"children\":[\"O(N log N + T)\",[\"$\",\"br\",null,{}],[\"$\",\"span\",null,{\"className\":\"text-muted-foreground\",\"children\":\"Retrieval + generation\"}]]}]]}],[\"$\",\"tr\",null,{\"children\":[[\"$\",\"td\",null,{\"className\":\"border border-border p-3 font-medium\",\"children\":[\"Generic Preference-Based\",[\"$\",\"br\",null,{}],[\"$\",\"span\",null,{\"className\":\"text-xs text-muted-foreground\",\"children\":\"Pareto, CP-Net, Utility\"}]]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-amber-600\",\"children\":\"â ï¸ Moderate\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-amber-600\",\"children\":\"â ï¸ Data gameable\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-green-600\",\"children\":\"â
Yes\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-green-600\",\"children\":\"â
Under model\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-red-600\",\"children\":\"â No\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-border p-3 text-xs\",\"children\":[\"NP-hard\",[\"$\",\"br\",null,{}],[\"$\",\"span\",null,{\"className\":\"text-muted-foreground\",\"children\":\"CP-net dominance testing\"}]]}]]}],[\"$\",\"tr\",null,{\"className\":\"bg-primary/10 border-2 border-primary\",\"children\":[[\"$\",\"td\",null,{\"className\":\"border border-primary p-3 font-bold text-primary\",\"children\":[\"Preference-DAG (US 2025/0298806)\",[\"$\",\"br\",null,{}],[\"$\",\"span\",null,{\"className\":\"text-xs font-normal\",\"children\":\"PerfectFinder\"}]]}],[\"$\",\"td\",null,{\"className\":\"border border-primary p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-green-600 font-semibold\",\"children\":\"â
Yes\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-primary p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-green-600 font-semibold\",\"children\":\"â
Yes\"}]}],[\"$\",\"td\",null,{\"className\":\"border border-primary p-3\",\"children\":[\"$\",\"span\",null,{\"className\":\"text-green-600 font-semibold\",\"children\":\"â
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