Multicriteria cuts and size-constrained k-cuts in hypergraphs

Calvin Beideman, Karthekeyan Chandrasekaran, Chao Xu

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We address counting and optimization variants of multicriteria global min-cut and size-constrained min-k-cut in hypergraphs. 1. For an r-rank n-vertex hypergraph endowed with t hyperedge-cost functions, we show that the number of multiobjective min-cuts is O(r2trn3t−1). In particular, this shows that the number of parametric min-cuts in constant rank hypergraphs for a constant number of criteria is strongly polynomial, thus resolving an open question by Aissi, Mahjoub, McCormick, and Queyranne [1]. In addition, we give randomized algorithms to enumerate all multiobjective min-cuts and all pareto-optimal cuts in strongly polynomial-time. 2. We also address node-budgeted multiobjective min-cuts: For an n-vertex hypergraph endowed with t vertex-weight functions, we show that the number of node-budgeted multiobjective min-cuts is O(r2rnt+2), where r is the rank of the hypergraph, and the number of node-budgeted b-multiobjective min-cuts for a fixed budget-vector b ∈ Rt+ is O(n2). 3. We show that min-k-cut in hypergraphs subject to constant lower bounds on part sizes is solvable in polynomial-time for constant k, thus resolving an open problem posed by Queyranne [11]. Our technique also shows that the number of optimal solutions is polynomial. All of our results build on the random contraction approach of Karger [12]. Our techniques illustrate the versatility of the random contraction approach to address counting and algorithmic problems concerning multiobjective min-cuts and size-constrained k-cuts in hypergraphs.

Original languageEnglish (US)
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2020
EditorsJaroslaw Byrka, Raghu Meka
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771641
DOIs
StatePublished - Aug 1 2020
Event23rd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 24th International Conference on Randomization and Computation, APPROX/RANDOM 2020 - Virtual, Online, United States
Duration: Aug 17 2020Aug 19 2020

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume176
ISSN (Print)1868-8969

Conference

Conference23rd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 24th International Conference on Randomization and Computation, APPROX/RANDOM 2020
CountryUnited States
CityVirtual, Online
Period8/17/208/19/20

Keywords

  • Hypergraph min-cut
  • Hypergraph-k-cut
  • Multiobjective Optimization

ASJC Scopus subject areas

  • Software

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