NK-MaxClique and MMCQ: Tow New Exact Branch and Bound Algorithms for the Maximum Clique Problem

Abstract

The maximum clique problem (MCP) is a fundamental problem in combinatorial optimization which finds important applications in real-word. This paper describes two new efficient branch-and-bound maximum clique algorithms NK-MaxClique and MMCQ, designed for solving MCP. We define some pruning conditions based on core numbers and vertex ordering to efficiently remove many of the search space. With respect to this ordering, the algorithms consider the vertices respectively to find the corresponding maximum clique in subproblems. Simulation results demonstrate that the algorithms outperform the previous wellknown algorithms for many instances when applied to DIMACS benchmark and random graphs.

NK-PMC: A new exact branch and bound parallel algorithm for the maximum clique problem

“…The core number of the members in N K(u) is at least core(u). Accordingly, the members in N K(u) may construct a clique of size core(u) + 1 (if it exists) [26]. Definition 2.…”

“…This process is iteratively done until the core number of all the vertices in G be computed. The matrix K n×n is known as the core-matrix [26] of G in which except when K uv = 1 for v ∈ N (u) that the core number of u is computed before the core number of v by Core-algorithm, for every u, v ∈ V we have K uv = 0. The core-matrix K can be computed by the Core-matrix algorithm in O(m) [26].…”

“…The matrix K n×n is known as the core-matrix [26] of G in which except when K uv = 1 for v ∈ N (u) that the core number of u is computed before the core number of v by Core-algorithm, for every u, v ∈ V we have K uv = 0. The core-matrix K can be computed by the Core-matrix algorithm in O(m) [26]. Degeneracy vertex-ordering of G is an ordering of the vertices such that each vertex has at most core max neighbors later than it in the ordering [9].…”

“…Definition 2. [26] The core number of u in G[N K(u)] is known as the local-N K core number of u which is denoted by LCN N K (u).…”

“…Since on many of the real instances the size of the N K sets are usually small enough, maximum cliques can be found in a reasonable time. The pruning conditions presented in [25] and [26] are used to decrease running time of the NK-PMC algorithm.…”