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 $k$ non-empty parts $(V_1, V_2, \ldots, V_k)$ -- 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 $n^{O(k^2)}p$-time deterministic algorithm, where $n$ 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).

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 ⊆ E 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 [17,25,28,39]. In contrast, it is only recently that polynomial-time algorithms for Hypergraph-k-Cut were developed [2,3,12]. The randomized polynomial-time algorithm for Hypergraph-k-Cut that was designed in 2018 [3] showed that the number of minimum k-cut-sets in a hypergraph is O(n2k–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 [2], 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 algorithm is 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.

We address counting and optimization variants of multicriteria global min-cut and size-constrained min-$k$-cut in hypergraphs. 1. For an $r$-rank $n$-vertex hypergraph endowed with $t$ hyperedge-cost functions, we show that the number of multiobjective min-cuts is $O(r2^{tr}n^{3t-1})$. In particular, this shows that the number of parametric min-cuts in constant rank hypergraphs for a constant number of criteria is strongly polynomial, thus resolving an open question by Aissi, Mahjoub, McCormick, and Queyranne (Math Programming, 2015). In addition, we give randomized algorithms to enumerate all multiobjective min-cuts and all pareto-optimal cuts in strongly polynomial-time. 2. We also address node-budgeted multiobjective min-cuts: For an $n$-vertex hypergraph endowed with $t$ vertex-weight functions, we show that the number of node-budgeted multiobjective min-cuts is $O(r2^{r}n^{t+2})$, where $r$ is the rank of the hypergraph, and the number of node-budgeted $b$-multiobjective min-cuts for a fixed budget-vector $b$ is $O(n^2)$. 3. We show that min-$k$-cut in hypergraphs subject to constant lower bounds on part sizes is solvable in polynomial-time for constant $k$, thus resolving an open problem posed by Queyranne. Our technique also shows that the number of optimal solutions is polynomial. All of our results build on the random contraction approach of Karger (SODA, 1993). Our techniques illustrate the versatility of the random contraction approach to address counting and algorithmic problems concerning multiobjective min-cuts and size-constrained $k$-cuts in hypergraphs.