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Greedy Maximal Independent Sets via Local Limits

Authors Michael Krivelevich , Tamás Mészáros , Peleg Michaeli , Clara Shikhelman

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Author Details

Michael Krivelevich
  • School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 6997801, Israel
Tamás Mészáros
  • Fachbereich Mathematik und Informatik, Institut für Mathematik, Freie Universität Berlin, 14195 Berlin, Germany
Peleg Michaeli
  • School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 6997801, Israel
Clara Shikhelman
  • Simons Institute for the Theory of Computing, Berkeley, CA, USA
  • School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 6997801, Israel

Funding

  • Krivelevich, Michael: Partially supported by USA - Israel BSF grants 2014361 and 2018267, and by ISF grant 1261/17.
  • Mészáros, Tamás: Research supported by the Dahlem Research School and by the Berlin Mathematics Research Center Math+.
  • Michaeli, Peleg: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement number 676970, RANDGEOM).

Acknowledgements

The authors wish to thank the organisers of the Joint FUB - TAU Workshop on Graph and Hypergraph Colouring, hosted by the Freie Universität Berlin in 2018, and Michal Amir, Shagnik Das, Lior Gishboliner, Matan Harel, Frank Mousset, Matan Shalev and Yinon Spinka for useful discussions and ideas.

Cite As Get BibTex

Michael Krivelevich, Tamás Mészáros, Peleg Michaeli, and Clara Shikhelman. Greedy Maximal Independent Sets via Local Limits. In 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 159, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.AofA.2020.20

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 - and 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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Random graphs
  • Mathematics of computing → Probabilistic algorithms
Keywords
  • Greedy maximal independent set
  • random graph
  • local limit

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