Deterministic enumeration of all minimum cut-sets and k-cut-sets in hypergraphs for fixed k
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We consider the problem of deterministically enumerating all minimum k-cut-sets in a given hypergraph for any fixed k. The input here is a hypergraph G = (V, E) with non-negative hyperedge costs. A subset F of hyperedges is a k-cut-set if the number of connected components in G - F is at least k and it is a minimum k-cut-set if it has the least cost among all k-cut-sets. For fixed k, we call the problem of finding a minimum k-cut-set as Hypergraph-k-Cut and the problem of enumerating all minimum k-cut-sets as Enum-Hypergraph-k-Cut. The special cases of Hypergraph-k-Cut and Enum-Hypergraph-k-Cut restricted to graph inputs are well-known to be solvable in (randomized as well as deterministic) polynomial time. In contrast, it is only recently that polynomial-time algorithms for Hypergraph-k-Cut were developed. The randomized polynomial-time algorithm for Hypergraph-k-Cut that was designed in 2018 (Chandrasekaran, Xu, and Yu, SODA 2018) showed that the number of minimum k-cut-sets in a hypergraph is O(n^2k-2), where n is the number of vertices in the input hypergraph, and that they can all be enumerated in randomized polynomial time, thus resolving Enum-Hypergraph-k-Cut in randomized polynomial time. A deterministic polynomial-time algorithm for Hypergraph-k-Cut was subsequently designed in 2020 (Chandrasekaran and Chekuri, FOCS 2020), but it is not guaranteed to enumerate all minimum k-cut-sets. In this work, we give the first deterministic polynomial-time algorithm to solve Enum-Hypergraph-k-Cut (this is non-trivial even for k = 2). Our algorithms are based on new structural results that allow for efficient recovery of all minimum k-cut-sets by solving minimum (S,T)-terminal cuts. Our techniques give new structural insights even for enumerating all minimum cut-sets (i.e., minimum 2-cut-sets) in a given hypergraph.
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