Combinatorial Dominance Guarantees for Heuristic Algorithms

An $f(n)$ $\textit{dominance bound}$ on a heuristic for some problem is a guarantee that the heuristic always returns a solution not worse than at least $f(n)$ solutions. In this paper, we analyze several heuristics for $\textit{Vertex Cover}$, $\textit{Set Cover}$, and $\textit{Knapsack}$ for dominance bounds. In particular, we show that the well-known $\textit{maximal matching}$ heuristic of $\textit{Vertex Cover}$ provides an excellent dominance bound. We introduce new general analysis techniques which apply to a wide range of problems and heuristics for this measure. Certain general results relating approximation ratio and combinatorial dominance guarantees for optimization problems over subsets are established. We prove certain limitations on the combinatorial dominance guarantees of polynomial-time approximation schemes (PTAS), and give inapproximability results for the problems above