Combinatorics of Local Search: An Optimal 4-Local Hall's Theorem for Planar Graphs

Authors Daniel Antunes, Claire Mathieu, Nabil H. Mustafa

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Daniel Antunes
Claire Mathieu
Nabil H. Mustafa

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Daniel Antunes, Claire Mathieu, and Nabil H. Mustafa. Combinatorics of Local Search: An Optimal 4-Local Hall's Theorem for Planar Graphs. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 8:1-8:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


Local search for combinatorial optimization problems is becoming a dominant algorithmic paradigm, with several papers using it to resolve long-standing open problems. In this paper, we prove the following `4-local' version of Hall's theorem for planar graphs: given a bipartite planar graph G = (B, R, E) such that |N(B')| >= |B'| for all |B'| <= 4, there exists a matching of size at least |B|/4 in G; furthermore this bound is tight. Besides immediately implying improved bounds for several problems studied in previous papers, we find this variant of Hall's theorem to be of independent interest in graph theory.
  • Planar graphs
  • Local search
  • Hall's theorem
  • Combinatorial optimization
  • Expansion


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