Distributed approximation of maximum independent set and maximum matching

We present a simple distributed Δ-approximation algorithm for maximum weight independent set (MaxIS) in the CONGEST model which completes in O(MIS(G) · log W) rounds, where Δ 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(log n log W) rounds, where n is the number of nodes. We also present a deterministic O(Δ + log∗ n)-round algorithm based on coloring. We then show how to use our Max IS 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 Δ/log log Δ) 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 Δ.