Recently, many studies have been devoted to finding \textit{diverse} solutions in classical combinatorial problems, such as \textsc{Vertex Cover} (Baste et al., IJCAI'20), \textsc{Matching} (Fomin et al., ISAAC'20) and \textsc{Spanning Tree} (Hanaka et al., AAAI'21). We initiate the algorithmic study of $k$-\textsc{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 $k$-\textsc{Diverse Minimum s-t Cuts} can be solved in strongly polynomial time for diversity measures (i) and (ii) via \textit{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 \textit{max-flow} computation. Our results stand in contrast to the problem of finding $k$ diverse \textit{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 $k$-\textsc{Diverse Minimum s-t Cuts} subject to diversity measure (iii) is NP-hard already for $k=3$.

翻译：近期，许多研究致力于在经典组合问题中寻找\textit{多样化}解，例如\textsc{点覆盖}（Baste等，IJCAI'20）、\textsc{匹配}（Fomin等，ISAAC'20）和\textsc{生成树}（Hanaka等，AAAI'21）。我们启动了$k$-\textsc{多样化最小s-t割集}的算法研究，该问题给定有向图$G = (V, E)$、两个指定顶点$s,t \in V$以及整数$k > 0$，要求找出$G$中具有最大多样性的$k$个最小$s$-$t$割集。我们针对最大化三种可应用于割集集合的多样性度量的复杂性进行了研究：(i) 所有成对汉明距离之和，(ii) 集合中割集的并集基数，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)，$k$-\textsc{多样化最小s-t割集}可通过\textit{子模函数最小化}在强多项式时间内求解。这一结果通过建立有序最小$s$-$t$割集集合与分配格理论之间的联系而获得。当问题局限于寻找互不相交的解集合时，我们提供了一种更实用的算法，该算法能找出最大规模的成对不相交最小$s$-$t$割集。对于最小$s$-$t$割规模较小的图，该算法的运行时间等同于单次\textit{最大流}计算所需时间。我们的结果与寻找$k$个多样化\textit{全局}最小割集问题形成鲜明对比——后者即使在不相交情形下也被证明是NP难的（Hanaka等，AAAI'23），并部分回答了Wagner（Networks，1990）关于改进寻找不相交最小$s$-$t$割集集合复杂性的长期开放问题。最后，我们证明在多样性度量(iii)下，$k$-\textsc{多样化最小s-t割集}即便在$k=3$时已是NP难的。