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Hypergraph k-cut in randomized polynomial time

  • Karthekeyan Chandrasekaran
  • Chao Xu
  • Xilin YuEmail author
Full Length Paper Series A
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Abstract

For a fixed integer \(k\ge 2\), the hypergraph k-cut problem asks for a smallest subset of hyperedges whose removal leads to at least k connected components in the remaining hypergraph. While graph k-cut is solvable efficiently (Goldschmidt and Hochbaum in Math. Oper. Res. 19(1):24–37, 1994), the complexity of hypergraph k-cut has been open. In this work, we present a randomized polynomial time algorithm to solve the hypergraph k-cut problem. Our algorithmic technique extends to solve the more general hedge k-cut problem when the subgraph induced by every hedge has a constant number of connected components. Our algorithm is based on random contractions akin to Karger’s min cut algorithm. Our main technical contribution is a non-uniform distribution over the hedges (hyperedges) so that random contraction of hedges (hyperedges) chosen from the distribution succeeds in returning an optimum solution with large probability. In addition, we present an alternative contraction based randomized polynomial time approximation scheme for hedge k-cut in arbitrary hedgegraphs (i.e., hedgegraphs whose hedges could have a large number of connected components). Our algorithm and analysis also lead to bounds on the number of optimal solutions to the respective problems.

Keywords

Hypergraph-k-cut Hedgegraph-k-cut Randomized algorithm 

Mathematics Subject Classification

90C27 Combinatorial Optimization 

Notes

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019

Authors and Affiliations

  1. 1.University of Illinois Urbana-ChampaignChampaignUSA
  2. 2.Yahoo! ResearchNew YorkUSA

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