Lower Bounds for Maximal Matchings and Maximal Independent Sets

There are distributed graph algorithms for finding maximal matchings and maximal independent sets in O(Δ + log^* n) communication rounds; here n is the number of nodes and Δ is the maximum degree. The lower bound by Linial (1987, 1992) shows that the dependency on n is optimal: these problems cannot...

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Bibliographic Details
Published in:Proceedings / annual Symposium on Foundations of Computer Science pp. 481 - 497
Main Authors: Balliu, Alkida, Brandt, Sebastian, Hirvonen, Juho, Olivetti, Dennis, Rabie, Mikael, Suomela, Jukka
Format: Conference Proceeding
Language:English
Published: IEEE 01.11.2019
Subjects:
Bipartite graph
Color
Complexity theory
Computational modeling
Distributed algorithms
Distributed computing
distributed graph algorithms
lower bounds
maximal independent set
Maximal matching
Upper bound
ISSN:2575-8454
Online Access:Get full text
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Summary:There are distributed graph algorithms for finding maximal matchings and maximal independent sets in O(Δ + log^* n) communication rounds; here n is the number of nodes and Δ is the maximum degree. The lower bound by Linial (1987, 1992) shows that the dependency on n is optimal: these problems cannot be solved in o(log^* n) rounds even if Δ = 2. However, the dependency on Δ is a long-standing open question, and there is currently an exponential gap between the upper and lower bounds. We prove that the upper bounds are tight. We show that maximal matchings and maximal independent sets cannot be found in o(Δ + log log n / log log log n) rounds with any randomized algorithm in the LOCAL model of distributed computing. As a corollary, it follows that there is no deterministic algorithm for maximal matchings or maximal independent sets that runs in o(Δ + log n / log log n) rounds; this is an improvement over prior lower bounds also as a function of n.
ISSN:2575-8454
DOI:10.1109/FOCS.2019.00037