MAX CUT in Weighted Random Intersection Graphs and Discrepancy of Sparse Random Set Systems

Abstract

Let V be a set of n vertices, {mathcal M} a set of m labels, and let {textbf{R}} be an m times n matrix ofs independent Bernoulli random variables with probability of success p; columns of {textbf{R}} are incidence vectors of label sets assigned to vertices. A random instance G(V, E, {textbf{R}}^T {textbf{R}}) of the weighted random intersection graph model is constructed by drawing an edge with weight equal to the number of common labels (namely [{textbf{R}}^T {textbf{R}}]_{v,u}) between any two vertices u, v for which this weight is strictly larger than 0. In this paper we study the average case analysis of Weighted Max Cut, assuming the input is a weighted random intersection graph, i.e. given G(V, E, {textbf{R}}^T {textbf{R}}) we wish to find a partition of V into two sets so that the total weight of the edges having exactly one endpoint in each set is maximized. In particular, we initially prove that the weight of a maximum cut of G(V, E, {textbf{R}}^T {textbf{R}}) is concentrated around its expected value, and then show that, when the number of labels is much smaller than the number of vertices (in particular, m=n^{alpha }, alpha <1), a random partition of the vertices achieves asymptotically optimal cut weight with high probability. Furthermore, in the case n=m and constant average degree (i.e. p = frac{Theta (1)}{n}), we show that with high probability, a majority type randomized algorithm outputs a cut with weight that is larger than the weight of a random cut by a multiplicative constant strictly larger than 1. Then, we formally prove a connection between the computational problem of finding a (weighted) maximum cut in G(V, E, {textbf{R}}^T {textbf{R}}) and the problem of finding a 2-coloring that achieves minimum discrepancy for a set system Sigma with incidence matrix {textbf{R}} (i.e. minimum imbalance over all sets in Sigma ). We exploit this connection by proposing a (weak) bipartization algorithm for the case m=n, p = frac{Theta (1)}{n} that, when it terminates, its output can be used to find a 2-coloring with minimum discrepancy in a set system with incidence matrix {textbf{R}}. In fact, with high probability, the latter 2-coloring corresponds to a bipartition with maximum cut-weight in G(V, E, {textbf{R}}^T {textbf{R}}). Finally, we prove that our (weak) bipartization algorithm terminates in polynomial time, with high probability, at least when p = frac{c}{n}, c<1.