Finding Diverse Minimum s-t Cuts

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 $k=3$.