Counting and Enumerating Optimum Cut Sets for Hypergraph k-Partitioning Problems for Fixed k

We consider the problem of enumerating optimal solutions for two hypergraph k-partitioning problems, namely, HYPERGRAPH-k-CUT and MINMAX-HYPERGRAPH-k-PARTITION. The input in hypergraph k-partitioning problems is a hypergraph G= (V,E) with positive hyperedge costs along with a fixed positive integer k. The goal is to find a partition of V into k nonempty parts (V₁,V₂,:::,V_k)—known as a k-partition—so as to minimize an objective of interest. (1) If the objective of interest is the maximum cut value of the parts, then the problem is known as MINMAX-HYPERGRAPH-k-PARTITION. A subset of hyperedges is a MINMAX-k-CUT-SET if it is the subset of hyperedges crossing an optimum k-partition for MINMAX-HYPERGRAPH-k-PARTITION. (2) If the objective of interest is the total cost of hyperedges crossing the k-partition, then the problem is known as HYPERGRAPH-k-CUT. A subset of hyperedges is a MIN-k-CUT-SET if it is the subset of hyperedges crossing an optimum k-partition for HYPERGRAPH-k-CUT. We give the first polynomial bound on the number of MINMAX-k-CUT-SETs and a polynomial-time algorithm to enumerate all of them in hypergraphs for every fixed k. Our technique is strong enough to also enable an n^O(k⁾p-time deterministic algorithm to enumerate all MIN-k-CUT-SETs in hypergraphs, thus improving on the previously known n^O(^k2)p-time deterministic algorithm, in which n is the number of vertices and p is the size of the hypergraph. The correctness analysis of our enumeration approach relies on a structural result that is a strong and unifying generalization of known structural results for HYPERGRAPH-k-CUT and MINMAX-HYPERGRAPH-k-PARTITION. We believe that our structural result is likely to be of independent interest in the theory of hypergraphs (and graphs).