The design and analysis of approximation algorithms for N P-hard problems is perhaps the most active research area in the theory of combinatorial algorithms. In this article, we study the notion of a combinatorial dominance guarantee as a way for assessing the performance of a given approximation algorithm. An f (n) dominance bound is a guarantee that the heuristic always returns a solution not worse than at least f (n) solutions. We give tight analysis of many heuristics, and establish novel and interesting dominance guarantees even for certain inapproximable problems and heuristic search algorithms. For example, we show that the maximal matching heuristic of VERTEX COVER offers a combinatorial dominance guarantee of 2 n − (1.839 + o(1)) n . We also give inapproximability results for most of the problems we discuss.
International audience
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.
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