Abstract
We study distributed algorithms for the maximum matching problem in the CONGEST model, where each message must be bounded in size. We give new deterministic upper bounds, and a new lower bound on the problem.
We begin by giving a distributed algorithm that computes an exact maximum (unweighted) matching in bipartite graphs, in O(n log n) rounds. Next, we give a distributed algorithm that approximates the fractional weighted maximum matching problem in general graphs. In a graph with maximum degree at most Delta, the algorithm computes a (1epsilon)approximation for the problem in time O(log(Delta W)/epsilon^2), where W is a bound on the ratio between the largest and the smallest edge weight. Next, we show a slightly improved and generalized version of the deterministic rounding algorithm of Fischer [DISC '17]. Given a fractional weighted maximum matching solution of value f for a given graph G, we show that in time O((log^2(Delta)+log^*n)/epsilon), the fractional solution can be turned into an integer solution of value at least (1epsilon)f for bipartite graphs and (1epsilon) * (g1)/g * f for general graphs, where g is the length of the shortest odd cycle of G. Together with the above fractional maximum matching algorithm, this implies a deterministic algorithm that computes a (1epsilon)* (g1)/gapproximation for the weighted maximum matching problem in time O(log(Delta W)/epsilon^2 + (log^2(Delta)+log^* n)/epsilon).
On the lowerbound front, we show that even for unweighted fractional maximum matching in bipartite graphs, computing an (1  O(1/sqrt{n}))approximate solution requires at least Omega~(D+sqrt{n}) rounds in CONGEST. This lower bound requires the introduction of a new 2party communication problem, for which we prove a tight lower bound.
BibTeX  Entry
@InProceedings{ahmadi_et_al:LIPIcs:2018:9795,
author = {Mohamad Ahmadi and Fabian Kuhn and Rotem Oshman},
title = {{Distributed Approximate Maximum Matching in the CONGEST Model}},
booktitle = {32nd International Symposium on Distributed Computing (DISC 2018)},
pages = {6:16:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770927},
ISSN = {18688969},
year = {2018},
volume = {121},
editor = {Ulrich Schmid and Josef Widder},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9795},
URN = {urn:nbn:de:0030drops97950},
doi = {10.4230/LIPIcs.DISC.2018.6},
annote = {Keywords: distributed graph algorithms, maximum matching, deterministic rounding, communication complexity}
}
Keywords: 

distributed graph algorithms, maximum matching, deterministic rounding, communication complexity 
Collection: 

32nd International Symposium on Distributed Computing (DISC 2018) 
Issue Date: 

2018 
Date of publication: 

04.10.2018 