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

Abstract

AbstractLet G(n,c/n) and Gr(n) be an n‐node sparse random graph and a sparse random r‐regular graph, respectively, and let I(n,r) and I(n,c) 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 supn I(n,3)/n ≤ .4591 with high probability (w.h.p.) as n → ∞, as shown in Frieze and Suen [Random Structures Algorithms 5 (1994), 649–664] and Bollabas [European J Combin 1 (1980), 311–316], respectively. In this paper we assume in addition that the nodes of the graph are equipped with nonnegative 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 limn I(n,c)/n can be computed exactly even when c > e, and limn I(n,r)/n can be computed exactly for some r ≥ 1. For example, when the weights are exponentially distributed with parameter 1, limn I(n,2e)/n ≈ .5517, and limn I(n,3)/n ≈ .6077. 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). For the case of exponential distributions, we compute the corresponding limits for every c > 0 and every r ≥ 2. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006