On the max‐cut of sparse random graphs

Abstract

We consider the problem of estimating the size of a maximum cut (Max-Cut problem) in a random Erdős-Rényi graph on n nodes and ⌊cn⌋ edges. It is shown in Coppersmith et al. [CGHS04] that the size of the maximum cut in this graph normalized by the number of nodes belongs to the asymptotic region [c/2 + 0.37613 √ c, c/2 + 0.58870 √ c] with high probability (w.h.p.) as n increases, for all sufficiently large c. The upper bound was obtained by application of the first moment method, and the lower bound was obtained by constructing algorithmically a cut which achieves the stated lower bound.In this paper we improve both upper and lower bounds by introducing a novel bounding technique. Specifically, we establish that the size of the maximum cut normalized by the number of nodes belongs to the interval [c/2 + 0.47523 √ c, c/2 + 0.55909 √ c] w.h.p. as n increases, for all sufficiently large c. Instead of considering the expected number of cuts achieving a particular value as is done in the application of the first moment method, we observe that every maximum size cut satisfies a certain local optimality property, and we compute the expected number of cuts with a given value satisfying this local optimality property. Estimating this expectation amounts to solving a rather involved two dimensional large deviations problem. We solve this underlying large deviation problem asymptotically as c increases and use it to obtain an improved upper bound on the Max-Cut value. The lower bound is obtained by application of the second moment method, coupled with the same local optimality constraint, and is shown to work up to the stated lower bound value c/2+0.47523 √ c. It is worth noting that both bounds are stronger than the ones obtained by standard first and second moment methods.Finally, we also obtain an improved lower bound of 1.36000n on the Max-Cut for the random cubic graph or any cubic graph with large girth, improving the previous best bound of 1.33773n.

On MAXCUT in strictly supercritical random graphs, and coloring of random graphs and random tournaments

“…They also established a similar result for the random regular graph model G n,r . Asymptotic bounds on MC(c) were obtained by Coppersmith et al (see [9]), Gamarnik and Li (see [21]) and Feige and Ofek (see [17]). All these bounds are of the form c…”

Local approximation of the Maximum Cut in regular graphs

“…On the other hand, Dembo, Montanari and Sen [7] showed that in random d-regular graphs, the maximum cut has size m · ( 1 2 + 0.763+o(1) √ d ) + o(m) with high probability, proving a conjecture of [10]. The existence of this constant ≈ 0.763 is also connected to a conjecture of Hatami, Lovász and Szegedy [15] on limits of sparse graphs (see also the conclusion of [26] where the conjecture is strongly disproved for maximum independent sets, improving on an earlier result of [11]).…”

“…This was improved by Lyons [20], who proved a lower bound of 0.89m for cubic graphs of girth at least 655. The best known lower bound for cubic graphs of large girth, 0.90m, was proved by Gamarnik and Li [10], using a result of Csóka, Gerencsér, Harangi, and Virág [5]. The bound of Lyons [20] holds for any d-regular graphs of large enough (but constant) girth: such graphs have a cut of size at least m · (…”

“…All the results mentioned above (except the result of Gamarnik and Li [10]) 1 can be translated into algorithms working in the CONGEST model in a constant number of rounds. In this model, each node of the graph corresponds to a processor with infinite computational power and has a unique ID (each ID is an integer between 1 and poly(n), where n denotes the number of vertices in the graph).…”

“…We would like to thank Jérémie Chalopin and Keren Censor-Hillel for their remarks on the complexity of finding an orientation using very small messages in the CONGEST model. We also thank Michal Dory for calling reference [6] to our attention, and David Gamarnik for pointing out references [5,7,10,20] to us.…”