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Brendan Ames(bpamesmath.uwaterloo.ca) Abstract: We consider the problems of finding a maximum clique in a graph and finding a maximumedge biclique in a bipartite graph. Both problems are NPhard. We write both problems as matrixrank minimization and then relax them using the nuclear norm. This technique, which may be regarded as a generalization of compressive sensing, has recently been shown to be an effective way to solve rank optimization problems. In the special cases that the input graph has a planted clique or biclique (i.e., a single large clique or biclique plus diversionary edges), our algorithm successfully provides an exact solution to the original instance. For each problem, we provide two analyses of when our algorithm succeeds. In the first analysis, the diversionary edges are placed by an adversary. In the second, they are placed at random. In the case of random edges for the planted clique problem, we obtain the same bound as Alon, Krivelevich and Sudakov as well as Feige and Krauthgamer, but we use different techniques. Keywords: clique,biclique,convex relaxation,randomized,nuclear norm,trace norm Category 1: Convex and Nonsmooth Optimization (Convex Optimization ) Category 2: Combinatorial Optimization (Other ) Citation: Download: [PDF] Entry Submitted: 01/21/2009 Modify/Update this entry  
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