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Limits RegistryMinimum vertex cover — finding the smallest set of vertices touching every edge of a graph — has a simple, well-known 2-approximation algorithm (via maximal matching or LP rounding), a result that has stood as essentially the best practical approach for decades. Khot and Regev's 2008 conditional hardness result shows that, assuming the Unique Games Conjecture, no polynomial-time algorithm can do better than a factor of 2 minus any fixed positive constant, meaning the simple 2-approximation is close to provably optimal, though a handful of very marginal unconditional improvements exist for special cases.
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Minimum vertex cover — finding the smallest set of vertices touching every edge of a graph — has a simple, well-known 2-approximation algorithm (via maximal matching or LP rounding), a result that has stood as essentially the best practical approach for decades. Khot and Regev's 2008 conditional hardness result shows that, assuming the Unique Games Conjecture, no polynomial-time algorithm can do better than a factor of 2 minus any fixed positive constant, meaning the simple 2-approximation is close to provably optimal, though a handful of very marginal unconditional improvements exist for special cases.
A simple greedy/LP-rounding algorithm achieves a 2-approximation for the minimum vertex cover problem on general graphs, and assuming the Unique Games Conjecture, no polynomial-time algorithm can achieve a (2 - epsilon)-approximation for any constant epsilon > 0, making 2 essentially optimal.
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Limits Registry. LR-VERTEX-COVER-APPROXIMATION. Best-known polynomial-time approximation ratio for minimum vertex cover. 2026.