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Finding maximum cliques in arbitrary and in special graphs

Bestimmung von Maximum Cliquen in beliebigen und in speziellen Graphen

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Abstract

The classical problem of finding a clique of largest cardinality in an arbitrary graph is NP-complete. For that reason earlier work diverges into two directions. The first concerns algorithms solving the problem for arbitrary graphs in reasonable (but exponential) time, the other restricts to special classes of graphs where polynomial methods can be found. Here, the two directions are combined in a way. A branch and bound algorithm is developed treating the general case. Computational experiments on random graphs show that this algorithm compares favorable to the fastest known method. Furthermore, it consumes only polynomial time for quite a few graph classes. For some of them no polynomial solution method is given so far.

Zusammenfassung

Das klassische Problem der Ermittlung einer Clique größter Mächtigkeit in einem beliebigen Graph ist NP-vollständig. Deshalb teilen sich bisherige Untersuchungen in zwei Richtungen. Die erste beschäftigt sich mit Algorithmen, die das Problem für beliebige Graphen in vernünftiger (aber exponentieller) Zeit lösen, die andere beschränkt sich auf spezielle Graphenklassen, für die polynomiale Methoden möglich sind. Hier werden diese beiden Richtungen kombiniert. Es wird ein Branch and Bound-Algorithmus für den allgemeinen Fall entwickelt. Praktische Rechenexperimente an Zufallsgraphen zeigen, daß dieser Algorithmus dem schnellsten bisher bekannten Verfahren überlegen ist. Darüberhinaus benötigt er nur polynomiale Rechenzeit für eine Vielzahl von Graphenklassen, darunter einige, für die noch keine polynomiale Lösungsmethode bekannt ist.

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  1. Mathematisches Institut, Technische Universität München, Arcisstraße 21, D-W-8000, München 2, Federal Republic of Germany

    L. Babel

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Babel, L. Finding maximum cliques in arbitrary and in special graphs. Computing 46, 321–341 (1991). https://doi.org/10.1007/BF02257777

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