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). Finding diverse solutions is important in settings where the user is not able to specify all criteria of the desired solution. Motivated by an application in the field of system identification, 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 two diversity measures for a collection of cuts: (i) the sum of all pairwise Hamming distances, and (ii) the cardinality of the union of cuts in the collection. We prove that $k$-DIVERSE MINIMUM $s$-$t$ CUTS can be solved in strongly polynomial time for both diversity measures 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. These 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., 2022) -- 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.

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，IJCAI'20）、匹配（Fomin等，ISAAC'20）和生成树（Hanaka等，AAAI'21）。在用户无法指定所需解的所有标准的情况下，寻找多样化解具有重要意义。受系统识别领域应用的启发，我们首次对$k$-多样最小$s$-$t$割问题进行算法研究：给定有向图$G = (V, E)$、两个指定顶点$s,t \in V$以及整数$k > 0$，要求寻找$G$中一组具有最大多样性的$k$个最小$s$-$t$割。针对割集的两种多样性度量，我们探究了该问题的复杂性：（i）所有成对汉明距离之和，（ii）割集中割的并集的基数。我们证明，对于这两种多样性度量，$k$-多样最小$s$-$t$割可通过次模函数最小化在强多项式时间内求解。这一结果是通过建立有序最小$s$-$t$割集与分配格理论之间的联系而获得的。当问题限制为仅寻找互不相交的解集时，我们提出了一种更实用的算法，该算法可找到一组最大规模的成对不相交最小$s$-$t$割。对于最小$s$-$t$割较小的图，该算法运行时间仅相当于一次最大流计算。这些结果与寻找$k$个多样全局最小割的问题形成鲜明对比——后者即使对于不相交情况也被证明是NP难的（Hanaka等，2022），并部分回答了Wagner（Networks 1990）关于改进寻找不相交最小$s$-$t$割集复杂性的长期开放问题。