A branch and bound algorithm for the maximum clique problem

Abstract

We present a branch and bound algorithm for the maximum clique problem in arbitrary graphs. The main part of the algorithm consists in the determination of upper bounds by graph colorings. Using a modification of a known graph coloring method called DSATUR we simultaneously derive iower and upper bounds for the clique number.Zusammenfassung: Wir stellen einen Branch and Bound Algorithmus fox dos Maximum Clique Problem in einem beliebigen Graphen vor. Das Hauptaugenmerk richter sich dabei auf die Bestimmung oberer Schranken mit Ililfe yon Fiirbungen yon Graphen. Es wird eine Modifikation einer bekannten FSrbungsmethode, genannt DSATUR, verwendet, mit der sich gleichzeitig obere und untere Schranken fOx die CliquezahI erstellen lassen.

Probabilistic Constructive Optimization Techniques

“…Finally, note that the closely related maximum clique problem also has a wide range of applications [24] and that numerous algorithms have been developed to locate the largest clique in the graph [25]. The MIS problem is equivalent to the maximum clique problem in the complemented graph [19].…”

An exact algorithm for the maximum stable set problem

“…Random graphs are the usual test problems for MSS algorithms and have been used by [2,3,4,9,11,26]. To assess the robustness of our algorithm with respect to the input data, we did a series of experiments with structured (and 'real life') problems, which are easy to reproduce.…”

“…At the first iteration (Fig. la) the algorithm produces a clique covering C 1 = {(1, 2), (3,4), (5)}. This gives UB 1 = 3, which is indeed the best possible value for an upper bound produced by any heuristic based on clique covering.…”

“…This is done by HEUTHETA in the following way. At the t-th iteration HEUTHETA finds a set of cliques C t such that each node is contained in exactly t cliques ofG t. The cliques ofC 1 are obtained by generating a clique covering of G and this can be done by any method (greedy or sophisticated as in [3,4,11]). …”

CABOB: A Fast Optimal Algorithm for Winner Determination in Combinatorial Auctions

“…Three fundamentally different approaches have been taken: (1) designing algorithms that provably find an optimal solution but are slow on some problem instances (Sandholm 2002a;Fujishima et al 1999;Sandholm and Suri 2003;Andersson et al 2000;de Vries and Vohra 2003;Gonen and Lehmann 2000;Lehmann and Gonen 2001;Leyton-Brown et al 2000b;van Hoesel and Müller 2001;Balas and Yu 1986;Babel and Tinhofer 1990;Babel 1991;Xue 1991, 1996;Nemhauser and Sigismondi 1992;Mannino and Sassano 1994;Pardalos and Desai 1991;Loukakis and Tsouros 1983), (2) designing algorithms that are provably fast but fail to find an optimal (or even close to optimal) solution to some problem instances (Lehmann et al 2002, Hoos and Boutilier 2000, Zurel and Nisan 2001, Anandalingam et al 2002, and (3) restricting the bundles on which bids can be submitted so severely that the remaining problem can be solved optimally and provably fast (Rothkopf et al 1998, Tennenholtz 2000, Penn and Tennenholtz 2000, van Hoesel and Müller 2001, Lehmann et al 2005.…”