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

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