## Abstract

We consider the HYPERGRAPH-k-CUT problem. The input consists of a hypergraph G = (V,E) with nonnegative hyperedge-costs c: E → R_{+} and a positive integer k. The objective is to find a minimum cost subset F ⊆ E such that the number of connected components in G – F is at least k. An alternative formulation of the objective is to find a partition of V into k nonempty sets V_{1},V_{2},:::,V_{k} so as to minimize the cost of the hyperedges that cross the partition. GRAPH-k-CUT, the special case of HYPERGRAPH-k-CUT obtained by restricting to graph inputs, has received considerable attention. Several different approaches lead to a polynomial-time algorithm for GRAPH-k-CUT when k is fixed, starting with the work of Goldschmidt and Hochbaum (Math of OR, 1994). In contrast, it is only recently that a randomized polynomial time algorithm for HYPERGRAPH-k-CUT was developed (Chandrasekaran, Xu, Yu, Math Programming, 2019) via a subtle generalization of Karger’s random contraction approach for graphs. In this work, we develop the first deterministic algorithm for HYPER-GRAPH-k-CUT that runs in polynomial time for any fixed k. We describe two algorithms both of which are based on a divide and conquer approach. The first algorithm is simpler and runs in n^{O}(^{k2)}m time while the second one runs in n^{O}(k^{)}m time, where n is the number of vertices and m is the number of hyperedges in the input hypergraph. Our proof relies on new structural results that allow for efficient recovery of the parts of an optimum k-partition by solving minimum (S,T)-terminal cuts. Our techniques give new insights even for GRAPH-k-CUT.

Original language | English (US) |
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Pages (from-to) | 3380-3399 |

Number of pages | 20 |

Journal | Mathematics of Operations Research |

Volume | 47 |

Issue number | 4 |

DOIs | |

State | Published - Nov 2022 |

## Keywords

- algorithms
- hypergraphs
- k-cut

## ASJC Scopus subject areas

- Mathematics(all)
- Computer Science Applications
- Management Science and Operations Research