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Hypergraph k-cut in randomized polynomial time

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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.

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Notes

  1. The node-weighted k-way cut problem is the following: Given a graph with weights on the nodes and a collection of terminal nodes, remove a smallest weight subset of non-terminal nodes so that the resulting graph has no path between the terminals.

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  1. Chao Xu

    Present address: Yahoo! Research, New York, USA

Authors and Affiliations

  1. University of Illinois Urbana-Champaign, Champaign, USA

    Karthekeyan Chandrasekaran, Chao Xu & Xilin Yu

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  1. Karthekeyan Chandrasekaran
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  2. Chao Xu
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Correspondence to Xilin Yu.

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Chao Xu and Xilin Yu were supported in part by NSF Grants CCF-1526799 and CCF-1319376 respectively. Karthekeyan is supported by NSF Grants CCF-1907937 and CCF-1814613.

An extended abstract of this work appeared in the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2018). The current version contains full proofs and cut counting results.

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Chandrasekaran, K., Xu, C. & Yu, X. Hypergraph k-cut in randomized polynomial time. Math. Program. 186, 85–113 (2021). https://doi.org/10.1007/s10107-019-01443-7

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