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 $c:E\to {\mathbb{R}}_{+}$ and a positive integer k. The objective is to find a minimum cost subset $F\subseteq E$ 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 $V_1,V_2,\dots ,V_k$ 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 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 $n^{O(k^2)}m$ 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.