Maximum weight independent sets and matchings in sparse random graphs exact results using the local weak convergence method

Abstract

Let G(n,c/n) and Gr(n) be an n-node sparse random and a sparse random r-regular graph, respectively, and let I(n, c) and I(n, r) be the sizes of the largest independent set in G(n,c/n) and Gr(n). The asymptotic value of 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 ≤ lim infn I(n, 3)/n ≤ lim supnI(n,3)/n ≤ .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 limnI(n,c)/n can be computed exactly even when c > e, and limn 2(n, r)/n can be computed exactly for some r > 2. For example, when the weights are exponentially distributed with parameter 1, limn I(n, 2e)/n ≈ .5517 in G(n,c/n), and lim n I(n, 3)/n ≈ .6077 in G3(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 Gr(n). © Springer-Verlag 2004.