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NC Algorithms for Weighted Planar Perfect Matching and Related Problems

Author Piotr Sankowski



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Piotr Sankowski
  • Institute of Informatics, University of Warsaw

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Piotr Sankowski. NC Algorithms for Weighted Planar Perfect Matching and Related Problems. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 97:1-97:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ICALP.2018.97

Abstract

Consider a planar graph G=(V,E) with polynomially bounded edge weight function w:E -> [0, poly(n)]. The main results of this paper are NC algorithms for finding minimum weight perfect matching in G. In order to solve this problems we develop a new relatively simple but versatile framework that is combinatorial in spirit. It handles the combinatorial structure of matchings directly and needs to only know weights of appropriately defined matchings from algebraic subroutines. Moreover, using novel planarity preserving reductions, we show how to find: maximum weight matching in G when G is bipartite; maximum multiple-source multiple-sink flow in G where c:E -> [1, poly(n)] is a polynomially bounded edge capacity function; minimum weight f-factor in G where f:V -> [1, poly(n)]; min-cost flow in G where c:E -> [1, poly(n)] is a polynomially bounded edge capacity function and b:V -> [1, poly(n)] is a polynomially bounded vertex demand function. There have been no known NC algorithms for these problems previously.

Subject Classification

ACM Subject Classification
  • Theory of computation → Network flows
  • Theory of computation → Shared memory algorithms
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • planar graph
  • NC algorithms
  • maximum cardinality matching
  • maximum weight matching
  • min-cost flow
  • maximum multiple-source multiple-sink flow
  • f-factors

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