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

Abstract

AbstractLet V be a set of n vertices, $${\mathcal M}$$ M a set of m labels, and let $${\textbf{R}}$$ R be an $$m \times n$$ m × n matrix ofs independent Bernoulli random variables with probability of success p; columns of $${\textbf{R}}$$ R are incidence vectors of label sets assigned to vertices. A random instance $$G(V, E, {\textbf{R}}^T {\textbf{R}})$$ G ( V , E , R T R ) of the weighted random intersection graph model is constructed by drawing an edge with weight equal to the number of common labels (namely $$[{\textbf{R}}^T {\textbf{R}}]_{v,u}$$ [ R T 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 $$G(V, E, {\textbf{R}}^T {\textbf{R}})$$ G ( V , E , R T 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 $$G(V, E, {\textbf{R}}^T {\textbf{R}})$$ G ( V , E , R T 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, $$m=n^{\alpha }, \alpha <1$$ m = n α , α < 1 ), a random partition of the vertices achieves asymptotically optimal cut weight with high probability. Furthermore, in the case $$n=m$$ n = m and constant average degree (i.e. $$p = \frac{\Theta (1)}{n}$$ p = Θ ( 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 $$G(V, E, {\textbf{R}}^T {\textbf{R}})$$ G ( V , E , R T R ) and the problem of finding a 2-coloring that achieves minimum discrepancy for a set system $$\Sigma $$ Σ with incidence matrix $${\textbf{R}}$$ R (i.e. minimum imbalance over all sets in $$\Sigma $$ Σ ). We exploit this connection by proposing a (weak) bipartization algorithm for the case $$m=n, p = \frac{\Theta (1)}{n}$$ m = n , p = Θ ( 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 $${\textbf{R}}$$ R . In fact, with high probability, the latter 2-coloring corresponds to a bipartition with maximum cut-weight in $$G(V, E, {\textbf{R}}^T {\textbf{R}})$$ G ( V , E , R T R ) . Finally, we prove that our (weak) bipartization algorithm terminates in polynomial time, with high probability, at least when $$p = \frac{c}{n}, c<1$$ p = c n , c < 1 .