Distributed Approximation of Maximum Independent Set and Maximum Matching

Abstract

We present a simple distributed ∆-approximation algorithm for maximum weight independent set (MaxIS) in the CONGEST model which completes in O(MIS(G) · log W ) 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(log n log W ) 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 ∆.

Hardness of Distributed Optimization

“…We note that there are many congest algorithms that are based on such simple operations, and in particular there are efficient O(log ∆)-approximations for MDS that are local aggregate algorithms. 3 We show that obtaining a better approximation requires different algorithms.…”

“…Lower bounds for local aggregate algorithms: We next use Lemma 4.7 to show lower bounds for MDS with restrictions on the algorithm. We start by defining the notion of a local aggregate algorithm, following the definition in [3] with slight changes. A function f is called orderinvariant if f (x 1 , ..., x n ) = f (x π(1) , ..., x π(n) ) for any permutation π.…”

Distributed Local Approximation Algorithms for Maximum Matching in Graphs and Hypergraphs

“…CONGEST 2 1/ǫ log n No [26] W.h.p. LOCAL ǫ −3 log n Yes [6] Det LOCAL ∆ 1/ǫ + ǫ −2 log * n No [6] Det LOCAL (log W n) 1/ǫ (∆ 1/ǫ + log * n) Yes [3] First-moment CONGEST 2 1/ǫ log ∆ log log ∆ No [8] Det CONGEST ∆ 1/ǫ + poly(1/ǫ) log * n No [11] Det LOCAL ǫ −9 log 5 ∆ log 2 n No [12] Our HMWM algorithms yields a (1 + ǫ)-approximation algorithm for graph matching: Theorem 1.3. For ǫ > 0, there is aÕ(ǫ −4 log 2 ∆ + ǫ −1 log * n)-round deterministic algorithm to get a (1 + ǫ)-approximate GMWM.…”

Beyond Alice and Bob: Improved Inapproximability for Maximum Independent Set in CONGEST

“…A maximum independent set in a (possibly weighted) graph is an independent set of maximum total weight, where by total we mean the sum of weights of nodes in the independent set. Independent sets play vital role in theoretical and practical computer science, and the problem of computing exact or approximate maximum independent set has been attracting attention recently in the CONGEST model [4,5,8,19]. However, in terms of upper bounds, we are still unable to find fast algorithms that achieve approximation factors better than Δ, where Δ is the maximum degree of a node in the graph.…”

“…However, in terms of upper bounds, we are still unable to find fast algorithms that achieve approximation factors better than Δ, where Δ is the maximum degree of a node in the graph. If one is happy with Δapproximation, or (1 + )Δ-approximation, then very fast and even sub-logarithmic algorithms exist [5,19]. In terms of lower bounds, recently Bachrach et al [4] built on the small-cut construction of [8], together with a very clever use of error-correcting codes, to prove a near-linear hardness for (5/6 + )-approximation, and near-quadratic one for (7/8 + )-approximation.…”