Detecting and characterizing small dense bipartite-like subgraphs by the bipartiteness ratio measure

Abstract

We study the problem of finding and characterizing subgraphs with small bipartiteness ratio. We give a bicriteria approximation algorithm SwpDB such that if there exists a subset S of volume at most k and bipartiteness ratio θ, then for any 0 < ε < 1/2, it finds a set S′ of volume at most 2k1+ε and bipartiteness ratio at most 4√θ/ε. By combining a truncation operation, we give a local algorithm LocDB, which has asymptotically the same approximation guarantee as the algorithm SwpDB on both the volume and bipartiteness ratio of the output set, and runs in time O(ε2 θ-2 k 1+ε ln 3 k), independent of the size of the graph. Finally, we give a spectral characterization of the small dense bipartite-like subgraphs by using the kth largest eigenvalue of the Laplacian of the graph. © 2013 Springer-Verlag.