Maximum Weight Independent Sets and Matchings in Sparse Random Graphs

Abstract

Let G(n,c/n) and G _r (n) be an n-node sparse random and a sparse random r-regular graph, respectively, and let ${\cal I}(n,c)$ and ${\cal I}(n,r)$ be the sizes of the largest independent set in G(n,c/n) and G _r (n). The asymptotic value of ${\cal I}(n,c)/n$ as n→∞, can be computed using the Karp-Sipser algorithm when c≤ e. For random cubic graphs, r=3, it is only known that $.432\leq\liminf_n {\cal I}(n,3)/n \leq \limsup_n {\cal I}(n,3)/n\leq .4591$ with high probability (w.h.p.) as n→∞, as shown in [FS94] and [Bol81], respectively.

In this paper we assume in addition that the nodes of the graph are equipped with non-negative weights, independently generated according to some common distribution, and we consider instead the maximum weight of an independent set. Surprisingly, we discover that for certain weight distributions, the limit $\lim_n {\cal I}(n,c)/n$ can be computed exactly even when c>e, and $\lim_n {\cal I}(n,r)/n$ can be computed exactly for some r≥ 2. For example, when the weights are exponentially distributed with parameter 1, $\lim_n {\cal I}(n,2e)/n\approx .5517$ in G(n,c/n), and $\lim_n {\cal I}(n,3)/n\approx .6077$ in G ₃(n). Our results are established using the recently developed local weak convergence method further reduced to a certain local optimality property exhibited by the models we consider. We extend our results to maximum weight matchings in G(n,c/n) and G _r (n).