Characterizing the Space of all Cliques in Random Graphs using "Go with the Winners"

We consider the problem of investigating the structure of the space of all possible cliques in random graphs generated according to the Gn; 12 ;K model. This model consists of all graphs on n nodes and edge probability 1=2, in which a random set ofK nodes is forced to be a clique. In this work we make a ̄rst attempt to explain the hardness of the problem by revealing the combinatorial characteristics of the space of all possible cliques for graphs generated according to the above distribution. Our main tool is \Go with the winners", an optimization heuristic that uses many particles that independently search the space of all possible solutions. In particular, we consider how the search space decomposes into smaller regions of related solutions by imposing a quality threshold to them. If these regions possess a combinatorial property, the so called \local expansion", then these regions can be e®ectively sampled by using enough particles and thus discover the optimal solution. Most importantly however, sampling can be used to deduce properties of the search space. These properties can then help optimize heuristic performance and design heuristics that take advantage of this information. Thus the goal of this work is not to compare cliquēnding heuristics but to exhibit a way to reveal the combinatorial characteristics of the search space, verify this information experimentally and use it to design good heuristics.