Recently, many studies have been devoted to finding diverse solutions in classical combinatorial problems, such as Vertex Cover (Baste et al., IJCAI'20), Matching (Fomin et al., ISAAC'20) and Spanning Tree (Hanaka et al., AAAI'21). We initiate the algorithmic study of $k$-Diverse Minimum s-t Cuts which, given a directed graph $G = (V, E)$, two specified vertices $s,t \in V$, and an integer $k > 0$, asks for a collection of $k$ minimum $s$-$t$ cuts in $G$ that has maximum diversity. We investigate the complexity of the problem for maximizing three diversity measures that can be applied to a collection of cuts: (i) the sum of all pairwise Hamming distances, (ii) the cardinality of the union of cuts in the collection, and (iii) the minimum pairwise Hamming distance. We prove that $k$-Diverse Minimum s-t Cuts can be solved in strongly polynomial time for diversity measures (i) and (ii) via submodular function minimization. We obtain this result by establishing a connection between ordered collections of minimum $s$-$t$ cuts and the theory of distributive lattices. When restricted to finding only collections of mutually disjoint solutions, we provide a more practical algorithm that finds a maximum set of pairwise disjoint minimum $s$-$t$ cuts. For graphs with small minimum $s$-$t$ cut, it runs in the time of a single max-flow computation. Our results stand in contrast to the problem of finding $k$ diverse global minimum cuts -- which is known to be NP-hard even for the disjoint case (Hanaka et al., AAAI'23) -- and partially answer a long-standing open question of Wagner (Networks, 1990) about improving the complexity of finding disjoint collections of minimum $s$-$t$ cuts. Lastly, we show that $k$-Diverse Minimum s-t Cuts subject to diversity measure (iii) is NP-hard already for $k=3$.

翻译：近年来，许多研究致力于在经典组合问题中寻找多样化解，例如顶点覆盖（Baste等人，IJCAI'20）、匹配（Fomin等人，ISAAC'20）和生成树（Hanaka等人，AAAI'21）。我们首次对$k$-多样化最小s-t割问题展开算法研究：给定有向图$G = (V, E)$、两个指定顶点$s,t \in V$及整数$k > 0$，要求找到$G$中具有最大多样性的$k$个最小$s$-$t$割集合。我们针对最大化三种适用于割集合的多样性度量研究了该问题的计算复杂度：（i）所有成对汉明距离之和，（ii）集合中割的并集基数，以及（iii）最小成对汉明距离。我们证明对于多样性度量（i）和（ii），$k$-多样化最小s-t割可通过次模函数最小化在强多项式时间内求解。该结果的获得是通过建立最小$s$-$t$割的有序集合与分配格理论之间的联系而实现的。当限制于寻找互不相交的解集合时，我们提出了一种更实用的算法来寻找最大规模的成对不相交最小$s$-$t$割集合。对于最小$s$-$t$割规模较小的图，该算法的运行时间仅相当于单次最大流计算。我们的结果与寻找$k$个多样化全局最小割的问题形成鲜明对比——后者即使在不相交情形下也被证明是NP难的（Hanaka等人，AAAI'23）——并部分回答了Wagner（Networks, 1990）关于改进寻找不相交最小$s$-$t$割集合复杂度的长期开放性问题。最后，我们证明对于多样性度量（iii），$k$-多样化最小s-t割问题在$k=3$时即已是NP难的。