Abstract
As is well known, the problem of finding a maximum clique in a graph is NP-hard. Nevertheless, NP-hard problems may have easy instances. This paper proposes a new, global optimization algorithm which tries to exploit favourable data constellations, focussing on the continuous problem formulation: maximize a quadratic form over the standard simplex. Some general connections of the latter problem with dynamic principles of evolutionary game theory are established. As an immediate consequence, one obtains a procedure which consists (a) of an iterative part similar to interior-path methods based on the so-called replicator dynamics; and (b) a routine to escape from inefficient, locally optimal solutions. For the special case of finding a maximum clique in a graph where the quadratic form arises from a regularization of the adjacence matrix, part (b), i.e. escaping from maximal cliques not of maximal size, is accomplished with block pivoting methods based on (large) independent sets, i.e. cliques of the complementary graph. A simulation study is included which indicates that the resulting procedure indeed has some merits.
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