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The Unique Games Conjecture and optimal approximation thresholds
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The Unique Games Conjecture and optimal approximation thresholds
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| The Unique Games Conjecture and optimal approximation thresholds. Proves Khot's Unique Games Conjecture. Independent direct reductions also establish NP-hardness, on unweighted graphs, of approximation beyond the Goemans–Williamson ratio for Max-Cut, below factor two for Vertex Cover, and within any fixed constant factor for Min-UnCut and directed feedback vertex set. These direct proofs use established PCP and Label Cover hardness results. |
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We prove the Unique Games Conjecture. For every fixed $\varepsilon,\delta\in(0,1/2)$, we give a deterministic polynomial-time reduction from 3SAT to Unique Games over a fixed finite alphabet, with completeness at least $1-\varepsilon$ and soundness at most δ.
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We prove that approximating Max-Cut on simple unweighted graphs within any fixed factor greater than the Goemans–Williamson constant is NP-hard.
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We prove that minimum Vertex Cover is NP-hard to approximate within every fixed factor below two, even on simple unweighted graphs.
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For every fixed C > 1, approximating Min-UnCut within factor C is NP-hard, even on simple undirected unweighted graphs.
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Approximating minimum directed feedback vertex set within any fixed constant factor is NP-hard, even on unweighted digraphs.
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