We present a simple distributed Δ-approximation algorithm for maximum
weight independent set (MaxIS) in the CONGEST model which completes
in O(MIS(G)⋅logW) 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. %Whether
our algorithm is randomized or deterministic depends on the \texttt{MIS}
algorithm used as a black-box.
Plugging in the best known algorithm for MIS gives a randomized solution in
O(lognlogW) 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 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 \emph{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(loglogΔlogΔ) 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