An Approximate Solution to the Minimum Vertex Cover Problem: The Salvador Algorithm

Source abstract

Minimum Vertex Cover has a classical polynomial-time ratio of 2, and a manuscript released by the OpenAI Math Team in October 2026, not yet peer reviewed, reports a proof of the Unique Games Conjecture in the form that, by the theorem of Khot and Regev, makes every fixed ratio below 2 NP-hard. We present SALVADOR, a worst-case O(n + m) ensemble of fourteen candidate covers and a swap-maximize refinement c15; it contains the Hvala ensemble, so it is valid and has ratio at most 2 on every graph. Two strategies are central. The weighted subdivision reduction c11 keeps the vertices of G, turns every edge into a node of a bipartite subdivision graph, and asks for a set of vertex-nodes that dominates every edge-node at minimum weight; it is solved by the exact degree-0, degree-1, triangle and degree-2 folding reductions, with a peel guided by the best earlier cover when no reduction applies. The reduce-and-search strategy c12 alternates a (1, 2)-swap local search with these reductions, the bounded-treewidth dynamic program c13 is exact when its elimination core is empty, and the consensus strategy c14 forces into the cover the vertices on which the earlier covers agree, as a self-loop would, and solves the rest of the graph again. We prove that every minimum-weight dominating selection is a minimum vertex cover, that c11 is a minimum cover whenever it needs no peel and otherwise exceeds the optimum by at most its number of peels, and that c12 and c14 never worsen their start. We also prove a bound for every graph: the returned cover has at most 2τ − r vertices, where r is the number of components with an edge. Hence the ratio is at most 1.999999999 on every graph with τ ≤ 109r, in particular on every graph with at most 109 + 1 vertices; the margin r/τ vanishes as τ grows, so this is not a (2 − ε)-approximation. Combined with per-class theorems, the ratio 1.999999999 is proved on every graph outside a residue of nearly K˝onig–Egerváry graphs with more than 109 vertices. In an ablation on the same instances, run with the quick mode of the released experiment scripts, c11 and c12 raise the number of optimal covers among 108 bipartite graphs from 97 to 105 and cut the total excess from 28 to 7 vertices, c12 cuts the excess on Khot–Regev YES instances with known optimum from 19 to 9 vertices, and c14 makes one more instance optimal where the others stop. On 206 triangle-free graphs, all 161 small ones are solved optimally, the ratio is at most 1.0016 on the 10 larger ones whose optimum was proved, and the certified ratio is at most 1.18 on Khot–Regev instances built on triangle-free constraint graphs. On the 68 DIMACS clique complements SALVADOR is never worse than Hvala, strictly better on 49 and optimal on 33, and on graphs of doubling size its running time grows with exponent 1.02 to 1.12 in n + m. If the reported hardness holds, a proof that SALVADOR achieves 2 − ε0 on all graphs would imply P = NP; we state this as a conditional theorem, not as a result.