cs.CC — Computational Complexity

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

Contributed by Frank Vega

Minimum Vertex Cover has a classical linear-time ratio of $2$, and under the Unique Games Conjecture no polynomial-time algorithm achieves a fixed ratio below $2$ unless $\mathrm{P}=\mathrm{NP}$. We present version 0.1.4 of \textsc{Salvador}, a linear-time ensemble of seventeen strategies, numbered in the order in which they are computed: twelve base strategies (component amplification before a maximal matching, the maximal matching itself, degree and weight heuristics, reductions to auxiliary problems that are solved exactly, a decomposition into bipartite layers, a weighted subdivision reduction and a bounded-treewidth dynamic program), followed by five improvers (a union pass, local search, reduce-and-search, consensus forcing and a swap-maximize refinement). We prove that every strategy runs in $O((n+m)\,\alpha(n))$ time, and that the returned cover has at most $2\tau-r$ vertices, where $r$ is the number of components with an edge, so the ratio is at most $1.999999999$ whenever $\tau\le10^{9}r$. We prove exactness or a bounded ratio on several classes of graphs, among them every bipartite graph on which the Hopcroft--Karp phases of the layer strategy reach a maximum matching, and we give a per-instance certificate: a cover with at most $2L-|M|$ vertices, for a lower bound $L\le\tau$ and the maximal matching $M$, is within ratio $3/2$ on that instance. In a reproducible suite the returned cover satisfies $|C|\le2\tau-|M|$ on all $291$ tested graphs with a known optimum and is certified on the six without one, although single strategies violate it; on the $68$ DIMACS clique complements it is never worse than Hvala and optimal on $37$ instances against $18$. Whether $|C|\le2\tau-|M|$ holds on every graph is stated as a conjecture: a proof would give a linear-time $3/2$-approximation and would refute the Unique Games Conjecture unless $\mathrm{P}=\mathrm{NP}$.