Greedy Maximal Independent Sets via Local Limits

Michael Krivelevich, Tamás Mészáros, Peleg Michaeli, Clara Shikhelman

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

The random greedy algorithm for finding a maximal independent set in a graph has been studied extensively in various settings in combinatorics, probability, computer science- A nd even in chemistry. The algorithm builds a maximal independent set by inspecting the vertices of the graph one at a time according to a random order, adding the current vertex to the independent set if it is not connected to any previously added vertex by an edge. In this paper we present a natural and general framework for calculating the asymptotics of the proportion of the yielded independent set for sequences of (possibly random) graphs, involving a useful notion of local convergence. We use this framework both to give short and simple proofs for results on previously studied families of graphs, such as paths and binomial random graphs, and to study new ones, such as random trees. We conclude our work by analysing the random greedy algorithm more closely when the base graph is a tree. We show that in expectation, the cardinality of a random greedy independent set in the path is no larger than that in any other tree of the same order. 2012 ACM Subject Classification Mathematics of computing Graph algorithms; Mathematics of computing Random graphs; Mathematics of computing Probabilistic algorithms.

Original languageEnglish
Title of host publication31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, A of A 2020
EditorsMichael Drmota, Clemens Heuberger
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771474
DOIs
StatePublished - 1 Jun 2020
Event31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, A of A 2020 - Klagenfurt, Austria
Duration: 15 Jun 202019 Jun 2020

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume159
ISSN (Print)1868-8969

Conference

Conference31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, A of A 2020
Country/TerritoryAustria
CityKlagenfurt
Period15/06/2019/06/20

Keywords

  • Greedy maximal independent set
  • Local limit
  • Random graph

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