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

The Minimum Vertex Cover problem is NP-hard. Its classical polynomial-time approximation ratio is 2, and Khot and Regev showed that, assuming the Unique Games Conjecture (UGC), no polynomial-time algorithm can achieve a factor \(2-\varepsilon\) for any fixed \(\varepsilon>0\). We present FindVertexCover, a linear-time ensemble algorithm that runs seven independently valid vertex-cover heuristics on the input graph \(G=(V,E)\) and returns the smallest cover found, then prunes redundant vertices. Four of the seven candidates—and the final redundant-vertex pruning pass applied to each of them—adapt a published ensemble that proves an unconditional \(O(n+m)\) time and space bound and a worst-case approximation ratio at most 2 for every graph, where \(n=|V|\) and \(m=|E|\). \textsc{Salvador} extends that ensemble with two further linear-time candidates of its own and a sixth candidate, SolveVC, that reduces \(G\) to a linear-size planar forest core via a weighted Minimum Independent Dominating Set (MIDS) gadget and solves that gadget with an accuracy-controlled Baker-style PTAS whose layering width is \(k=\lceil 1/\varepsilon\rceil\); non-core edges are covered by a greedy repair step. At the package default \(\varepsilon=1\), the Baker layering width is \(k=1\) and the PTAS pass degenerates to its linear-time greedy baseline, so every one of the seven candidates—and hence the ensemble as a whole—runs in worst-case \(O(n+m)\) time; because one candidate is the classical maximal-matching heuristic, the ensemble unconditionally achieves approximation ratio at most 2 on every graph, a theorem rather than a conjecture. We conjecture a universal ratio bound \(R^\star<2\) for the default call; on a reproducible suite of 1,746 test graphs, ranging from exhaustively enumerated small graphs to random and adversarial graphs with up to 200,000 vertices, the observed ratio never exceeds $1.61$, well below the classical \(7/4\) mark. If a bound below 2 were proved for the ensemble, then, under the standard assumption \(\mathrm{P}\neq\mathrm{NP}\), the Khot–Regev UGC-based hardness theorem for Vertex Cover would force the Unique Games Conjecture to be false. An open-source implementation is released as the Salvador package (v0.0.7).