Hypergraph k-Cut for Fixed k in Deterministic Polynomial Time

• 成果类型: Article; Early Access
• 署名作者: Chandrasekaran, Karthekeyan; Chekuri, Chandra
• 署名单位: University of Illinois System; University of Illinois Urbana-Champaign
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2021.1250
• 发表日期: 2022
• 关键词: algorithm

摘要

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 V1, V2, ..., Vk 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 HYPERGRAPH-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 nO(k2)m time while the second one runs in nO(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.