- The design and analysis of approximation algorithms for NP -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.