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 non-negative hyperedge-costs $c: E\rightarrow R_+$ 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). In contrast, it is only recently that a randomized polynomial time algorithm for Hypergraph-$k$-cut was developed (Chandrasekaran, Xu, Yu, 2018) 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.