The.Algorithm.Design.Manual.Springer-Verlag.1998

Clique
Heuristics for finding large cliques based on randomized techniques such as simulated annealing are
likely to work reasonably well.
Implementations: Programs for the closely related problems of finding cliques and independent sets
were sought for the Second DIMACS Implementation Challenge, held in October 1993. Programs and
data from the challenge are available by anonymous ftp from dimacs.rutgers.edu. Source codes are
available under pub/challenge/graph and test data under pub/djs.
In particular, two C language programs by David S. Johnson and David L. Applegate are available. The
dfmax.c implements a simple-minded branch-and-bound algorithm similar to that of [CP90]. The
dmclique.c uses a ``semi-exhaustive greedy'' scheme for finding large independent sets described in
[JAMS91]. Performance data for both programs is available in files results.dfmax and results.dmclique in
directories /pub/challenge/graph/benchmarks/clique and /pub/challenge/graph/benchmarks/volume.
Combinatorica [Ski90] provides (slow) Mathematica implementations of cliques, independent sets, and
vertex covers. See Section .
Notes: Good expositions of the proof that clique is NP-complete [Kar72] include [CLR90, GJ79,
Man89]. It is also given in Section . This reduction established that clique, vertex cover, and
independent set are very closely related problems, so heuristics and programs that solve one of them may
also produce reasonable solutions for the other two.
The linear-time algorithm for constructing maximal induced subgraphs is discussed in [Man89]. That
clique cannot be approximated to within a factor of is shown in [BGS95].
Related Problems: Independent set (see page ), vertex cover (see page ).
Next: Independent Set Up: Graph Problems: Hard Problems Previous: Graph Problems: Hard Problems
Algorithms
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