Hypergraph k-Cut for Fixed k in Deterministic Polynomial Time

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,:::,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.