Hypergraph-cut for fixed 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 rightarrow mathbb{R}+ and a positive integer k. The objective is to find a least-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 non-empty sets V {1}, V {2}, ldots, 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 (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 n{O(k{2})} time while the second one runs in n{O(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.