Branch and bound algorithms for the maximum clique problem under a unified framework

Abstract

In this paper we review branch and bound-based algorithms proposed for the exact solution of the maximum clique problem and describe them under a unifying conceptual framework. As a proof of concept, we actually implemented eight of these algorithms as parametrized versions of one single general branch and bound algorithm. The purpose of the present work is double folded. In the one hand, the implementation of several different algorithms under the same computational environment allows for a more precise assessment of their comparative performance at the experimental level. On the other hand we see the unifying conceptual framework provided by such description as a valuable step toward a more fine grained analysis of these algorithms.

A review on algorithms for maximum clique problems

“…From these results, one finds that the most effective algorithms are those that use vertex coloring for both bounding and branching like MCQ (Tomita & Seki, 2003), MCR (Tomita & Kameda, 2007), MCS (Tomita, Sutani, Higashi, Takahashi, & Wakatsuki, 2010), MaxCliqueDyn (Konc & Janežič, 2007) and BB-MaxClique (Segundo, Rodríguez-Losada, & Jiménez, 2011), and the MaxSAT based algorithms like MaxCLQ (Li & Quan, 2010). In Carmo & Züge (2012); Li & Quan (2010) and Tomita & Seki (2003), extensive comparisons are presented between the coloring based algorithm MCQ and some other algorithms before MCQ, like Cliquer (Östergård, 2002) and χ + DF (Fahle, 2002), disclosing that MCQ is superior on a wide range of DIMACS instances. Table 3: Performance comparison of 10 stat-of-art exact MCP algorithms on 35 popular DIMACS instances.…”

Bag of frequent subgraphs approach for image classification

“…it is a subset S of vertices with every two vertices in S forming an edge in G. A maximum clique is a clique that includes the largest possible number of vertices. The maximum clique problem [24] is NP-complete. The time complexity of the maximum clique detection [24] is known as O(2 n ) with n as the number of vertices in the graph.…”

“…The maximum clique problem [24] is NP-complete. The time complexity of the maximum clique detection [24] is known as O(2 n ) with n as the number of vertices in the graph. Fortunately, maximum clique testing algorithms [24,25] have been developed, making the maximum clique detection problem affordable in practice.…”

“…The time complexity of the maximum clique detection [24] is known as O(2 n ) with n as the number of vertices in the graph. Fortunately, maximum clique testing algorithms [24,25] have been developed, making the maximum clique detection problem affordable in practice. Most algorithms [24][25][26] for solving the maximum clique detection problem employ a pruning approach that reduces the search space by discarding vertices that are not able to produce cliques larger than the current maximum clique.…”

“…For the maximum clique detection, we have used the MaxCliqueDyn method presented in [25]. As reported in [24,25,32], the MaxCliqueDyn method is considerably faster than many other maximum clique algorithms. We present in Fig.…”