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). 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.