TY - GEN

T1 - Greedy Maximal Independent Sets via Local Limits

AU - Krivelevich, Michael

AU - Mészáros, Tamás

AU - Michaeli, Peleg

AU - Shikhelman, Clara

N1 - Publisher Copyright:
© 2020 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH Dagstuhl Publishing. All rights reserved.

PY - 2020/6/1

Y1 - 2020/6/1

N2 - 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.

AB - 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.

KW - Greedy maximal independent set

KW - Local limit

KW - Random graph

UR - http://www.scopus.com/inward/record.url?scp=85091025846&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.AofA.2020.20

DO - 10.4230/LIPIcs.AofA.2020.20

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AN - SCOPUS:85091025846

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, A of A 2020

A2 - Drmota, Michael

A2 - Heuberger, Clemens

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

Y2 - 15 June 2020 through 19 June 2020

ER -