Hypergraph $k$-cut for fixed $k$ in deterministic polynomial time

Karthekeyan Chandrasekaran, Chandra Chekuri

Abstract: We consider the Hypergraph- k-Cut problem. The input consists of a hypergraph G = (V, E) with nonnegative hyperedge-costs c:E→ \mathbbR+ and a positive integer k. The objective is to find a least-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 non-empty 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 (1988) [1], [2]. In contrast, it is only recently that a randomized polynomial time algorithm for Hypergraph- k-Cut was developed [3] via a subtle generalization of Karger's random contraction approach for graphs. In this work, we develop the first deterministic polynomial time algorithm for Hypergraph- k-Cut for all 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) time while the second one runs in nO(k) time. 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.