Distributed Approximation of Maximum Independent Set and Maximum Matching

We present a simple distributed ∆-approximation algorithm for maximum weight independent set (MaxIS) in the CONGEST model which completes in O(MIS(G) · logW ) rounds, where ∆ is the maximum degree, MIS(G) is the number of rounds needed to compute a maximal independent set (MIS) on G, and W is the maximum weight of a node. Plugging in the best known algorithm for MIS gives a randomized solution in O(logn logW ) rounds, where n is the number of nodes. We also present a deterministic O(∆ + log∗ n)-round algorithm based on coloring. We then show how to use our MaxIS approximation algorithms to compute a 2-approximation for maximum weight matching without incurring any additional round penalty in the CONGEST model. We use a known reduction for simulating algorithms on the line graph while incurring congestion, but we show our algorithm is part of a broad family of local aggregation algorithms for which we describe a mechanism that allows the simulation to run in the CONGEST model without an additional overhead. Next, we show that for maximum weight matching, relaxing the approximation factor to (2 + ε) allows us to devise a distributed algorithm requiring O( log ∆ log log ∆ ) rounds for any constant ε > 0. For the unweighted case, we can even obtain a (1+ε)-approximation in this number of rounds. These algorithms are the first to achieve the provably optimal round complexity with respect to dependency on ∆. ∗Technion, Department of Computer Science, {reuven, ckeren}@cs.technion.ac.il, [email protected]. Supported in part by the Israel Science Foundation (grant 1696/14). †ETH Zurich, [email protected]. ar X iv :1 70 8. 00 27 6v 1 [ cs .D C ] 1 A ug 2 01 7