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).

翻译：我们考虑两个超图 $k$ 划分问题（即 Hypergraph-$k$-Cut 和 Minmax-Hypergraph-$k$-Partition）的最优解枚举问题。超图 $k$ 划分问题的输入是一个具有正超边成本的超图 $G=(V, E)$，以及一个固定的正整数 $k$。目标是找到 $V$ 的一个划分，将其分成 $k$ 个非空部分 $(V_1, V_2, \ldots, V_k)$（称为 $k$ 划分），以最小化某个目标函数。1. 若目标函数是各部分的最大割值，则该问题称为 Minmax-Hypergraph-$k$-Partition。若一个超边子集是 Minmax-Hypergraph-$k$-Partition 的最优 $k$ 划分所跨越的超边子集，则称该超边子集为 minmax-$k$-割集。2. 若目标函数是跨越 $k$ 划分的超边总成本，则该问题称为 Hypergraph-$k$-Cut。若一个超边子集是 Hypergraph-$k$-Cut 的最优 $k$ 划分所跨越的超边子集，则称该超边子集为 min-$k$-割集。我们给出了 minmax-$k$-割集数量的首个多项式界，并给出了一个多项式时间算法，用于枚举所有固定 $k$ 的超图中的这些割集。我们的方法足够强大，还能得到一个 $n^{O(k)}p$ 时间的确定性算法，用于枚举超图中的所有 min-$k$-割集，从而改进了先前已知的 $n^{O(k^2)}p$ 时间确定性算法，其中 $n$ 是顶点数，$p$ 是超图的大小。我们枚举方法的正确性分析依赖于一个结构性结果，该结果是对 Hypergraph-$k$-Cut 和 Minmax-Hypergraph-$k$-Partition 已知结构性结果的强有力且统一的推广。我们相信我们的结构性结果在超图（和图）理论中可能具有独立的意义。