Veranstaltungen

The densest subgraph problem in a graph (DSG), in the simplest form, is the following. Given an undirected graph G = (V, E) find a subset S of vertices that maximizes the ratio |E(S)|/|S| where E(S) is the set of edges with both endpoints in S. DSG and several of its variants are well-studied in theory and practice and have several applications in data mining and network analysis. We study fast algorithms and structural aspects of DSG via the lens of supermodularity. For this we consider the densest supermodular subset problem (DSS): given a non-negative supermodular function f over a ground set V, maximize f(S)/|S|. Greedy peeling algorithms have been very popular for DSG and several variants due to their efficiency, empirical performance, and worst-case approximation guarantees. We describe a simple peeling algorithm for DSS and analyze its approximation guarantee in a fashion that unifies several existing results. Boob et al. developed an iterative peeling algorithm for DSG which appears to work very well in practice, and made a conjecture about its convergence to optimality. We affirmatively answer their conjecture, and in fact prove that a natural generalization of their algorithm converges for DSS. In subsequent work we also showed that the algorithm converges to the densest supermodular set decomposition. The proofs are based on connections between the LP relaxation for DSG suggested by Charikar, Lovász extension of a supermodular function, Fujishige's result on lexicographically optimal base in a (contra) polymatroid, and continuous optimization algorithms such Frank-Wolfe method and Multiplicative Weight Updates. Based on joint work in papers with Harb El Farouk, Kent Quanrud and Manuel Torres.