Finding Diverse Minimum s-t Cuts

TL;DR

It is proved that $k-Diverse Minimum s-t Cuts can be solved in strongly polynomial time for diversity measures (i) and (ii) via submodular function minimization, and it is shown that $k$-Diverse Minimum s-t Cuts subject to diversity measure (iii) is NP-hard already for $k=3$.

Abstract

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 $k=3$.

Figures (6)

• Figure 1: Example of Birkhoff's representation theorem for distributive lattices. The left is a distributive lattice $L$, the middle is the isomorphic lattice $\mathcal{D}(J(L))$ of ideals of join-irreducibles of $L$, and the right shows the directed graph $G(L)$ representing $L$. The join irreducible elements of $L$ and $\mathcal{D}(J(L))$ are highlighted in blue.
• Figure 2: Example of an $s$-$t$ path graph of height 4. Edges are labeled by integers corresponding to the path they belong to. Path edges are drawn in black and non-path edges in gray.
• Figure 3: Example illustrating the notions of crossing and invalid edges for an $s$-$t$ mincut $X$. Path edges are drawn in black and non-path edges in gray. Edges $e, f \in X$ are highlighted in blue. The edge $g$ is invalid with respect to $X$ since the edge $h$ is crossing with respect to it.
• Figure 4: Example illustrating the first two iterations of Algorithm \ref{['algo.kdisjointMSP']} on a path graph of height 4. The black- and gray-shaded vertices represent vertices marked at the previous and current iterations, respectively. The red edges correspond to the $s$-$t$ mincut found at the end of the first (left) iteration. Similarly, the blue edges correspond to the $s$-$t$ mincut found at the second (right) iteration.
• Figure 5: An instance $(G, X, Y, \ell)$ of 2-Fixed 3-DMC (left) and the constructed instance $(H, s, t, \ell')$ of Min-$3$-DMC (right). A solution for 2-Fixed 3-DMC in $G$ with $\ell = m/2$ can be mapped to a solution for Min-$3$-DMC in $H$ with $\ell' = m/2$ and vice versa.

Theorems & Definitions (57)

• theorem 1.1
• theorem 1.2
• theorem 1.3
• theorem 2.1: Birkhoff's Representation Theorem birkhoff1937rings
• theorem 2.2: murota2003 and markowsky2001overview
• proposition 3.0
• corollary 3.1
• lemma 3.2: escalante1972schnittverbande
• lemma 3.3
• proof