Experimental Evaluation of Approximation Algorithms for Maximum Distance-Bounded Subgraph Problems

Abstract

In this paper we consider two distance-based relaxed variants of the maximum clique problem (MAX CLIQUE), named MAX d-CLIQUE and MAX d-CLUB: A d-clique in a graph G is a subset S ⊆ V(G) of vertices such that for pairs of vertices u, v e S, the distance between u and v is at most d in G. A d-club in a graph G is a subset S' ⊆ V(G) of vertices that induces a subgraph of G of diameter at most d. MAX d-CLIQUE and MAX d-CLUB ask to find a maximum d-clique and a maximum d-club in a given unweighted graph, respectively. MAX 1-CLIQUE and MAX 1-CLUB cannot be efficiently approximated within a factor of n1-e for any e > 0 unless P = NP since they are identical to MAX CLIQUE [1], [2]. Also, it is known [3], [4] that it is NP-hard to approximate MAX d-CLIQUE and MAX d-CLUB to within a factor of n1/2–e for any fixed d ≥ 2 and any e > 0. As for approximability of MAX d-CLIQUE and MAX d-CLUB, [3] proposes a polynomial-time algorithm, called ByFindStard, and proves that its approximation ratio is O(n1/2) and O(n2/3) for any even d ≥ 2 and for any odd d ≥ 3, respectively. Very recently, a polynomial-time algorithm, called ByFindStar2d, achieving an optimal approximation ratio of O(n1/2) for MAX d-CLIQUE and MAX d-CLUB is designed for any odd d ≥ 3 in [4]. In this paper we implement those approximation algorithms and evaluate their quality empirically for random graphs Gn,p, which have n vertices and each edge appears with probability p. The experimental results show that (i) ByFindStar2d of approximation ratio O(n1/2) can find larger d-clubs (d-cliques) than ByFindStard of approximation ratio O(n2/3) for odd d, (ii) the size of d-clubs (d-cliques) output by ByFindStard is the same as ones by ByFindStar2d for even d, and (iii) ByFindStard can find the same size of d-clubs (d-cliques) much faster than ByFindStar2d.