An Efficient Branch-and-Bound Algorithm for Finding a Maximum Clique

Abstract

Abstract. We present an exact and efficient branch-and-bound algorithm for finding a maximum clique in an arbitrary graph. The algorithm is not specialized for any particular kind of graph. It employs approximate coloring and appropriate sorting of vertices to get an upper bound on the size of a maximum clique. We demonstrate by computational experiments on random graphs with up to 15,000 vertices and on DIMACS benchmark graphs that our algorithm remarkably outperforms other existing algorithms in general. It has been successfully applied to interesting problems in bioinformatics, image processing, the design of quantum circuits, and the design of DNA and RNA sequences for bio-molecular computation.

Finding Significant Web Pages with Lower Ranks by Pseudo-Clique Search

“…Upper bounds for the maximum clique size have been widely utilized in efficient depth-first branch-and-bound algorithms for finding maximum cliques [4,5,9]. The literature [5] has argued that the (vertex) chromatic number χ can provide the tightest upper bound.…”

“…However, identifying χ is an N P -complete problem. Therefore, approximations of χ are usually computed [4,5,9].…”

An improved bit parallel exact maximum clique algorithm

“…One of the most successful paradigms for fast MCP algorithms is branch-andbound, where a systematic enumeration of maximal diques is pruned by bounding the size of the largest possible clique in the remaining subproblem. A good compromise between computational overhead and tight bounding is obtained through approximate sequential vertex-coloring (usually referred to as SEQ), as in [7][8][9][10][11], and two recent leading algorithms MCS [12] and BB-MaxClique [13].…”

“…Reported tests clearly validate the combined improvements w.r.t. a previous algorithm MCQ [10] (and other state of the art algorithms) but the concrete contribution of recoloring is unfortunately not made explicit.…”

Iterated k-Opt Local Search for the Maximum Clique Problem

“…Several exact methods such as branch-and-bound algorithms [23,20] have been proposed to solve the MCP exactly. However, their effectiveness and applicability are limited to relatively small or sparse graphs.…”