Combinatorial dominance guarantees for problems with infeasible solutions

Abstract

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.

Average value of solutions for the bipartite boolean quadratic programs and rounding algorithms

“…Similar analysis for the three-dimensional assignment problem was considered by Sarvanov [31], for the Maximum clique problem by Bendall and Margot [6], and for the satisfiability problem by Twitto [36]. Berend et al [7] considered dominance analysis by including infeasible solutions. Other problems studied from the point of view of dominance analysis and average value based analysis include graph bipartition, variations of maximum clique and independent set problems [5,14] and the subset-sum problem [7].…”

“…Such a solution has interesting domination properties and hence the approach is also relevant in dominance analysis of heuristics. For recent developments on domination analysis, we refer to the excellent research papers [2,7,11,15]. Gutin and Yeo [16], Sarvanov [31], and Angel et al [5] studied heuristics for the quadratic assignment problem with performance guarantee in terms of average value of solutions.…”

Worst case analysis of Max-Regret, Greedy and other heuristics for Multidimensional Assignment and Traveling Salesman Problems

“…The fact that the greedy algorithm is of domination number 1 for s-AP (s ≥ 3) as well (see Theorem 3.2) implies that the algorithm should be used with great care for s-AP. Bounds for domination numbers/ratios were obtained for many CO heuristics; see, e.g., Berend et al (2007), Gutin and Yeo (2005), Koller and Noble (2004), and Punnen et al (2003).…”

A domination algorithm for {0,1}‐instances of the travelling salesman problem

“…For example, they gave a ‐domination algorithm for the minimum partition problem and an ‐domination algorithm for the max‐cut problem. Berend, Skiena and Twitto introduced a notion of combinatorial dominance for constrained problems, where not all permutations, subsets or partitions form feasible solutions. They analysed various algorithms for the maximum clique problem, the minimum subset cover and the maximum subset sum problem with respect to their notion of combinatorial dominance.…”