An FPT Algorithm for Diverse Minimum s-t Cuts

2026-07-03

AI Summary

This study addresses the problem of finding $k$ pairwise minimum $s$-$t$ cuts in a directed weighted graph such that the symmetric difference between any two cuts is at least $d$. The problem is NP-complete for $k \geq 3$ and closely related to a #P-complete counting problem. The authors uncover, for the first time, structural properties of diverse minimum $s$-$t$ cuts and nontrivially adapt flow augmentation techniques to this setting. Leveraging these insights, they design a fixed-parameter tractable (FPT) algorithm parameterized by $k + d$, thereby overcoming previous complexity barriers and enabling efficient parameterized computation of diverse minimum cuts.

Abstract

We study the problem of finding a family of diverse minimum edge s-t cuts in a directed weighted graph G. Given integers k and d, the task is to decide whether G contains k minimum s-t cuts C_1, ..., C_k such that for any i,j in [k], the number of edges in the symmetric difference of C_i and C_j is at least d. For d being 1 or 2, the problem corresponds to counting minimum s-t cuts in G, which is #P-complete [Provan and Ball, SICOMP 1983]. The problem is also known to be NP-complete already for k = 3 [de Berg, López Martínez, Spieksma, ISAAC 2024]. Our main result shows that the problem is fixed-parameter tractable (FPT) when parameterized by the combined parameter k + d. The main ingredients of our FPT algorithm build on novel structural properties of diverse minimum s-t cuts and a non-trivial application of the flow-augmentation technique of Kim, Kratsch, Pilipczuk, and Wahlström [JACM 2025].

Problem

• diverse minimum s-t cuts
• symmetric difference
• fixed-parameter tractability
• NP-complete
• edge cuts

Innovation

• fixed-parameter tractability
• diverse minimum s-t cuts
• flow augmentation
• symmetric difference
• parameterized algorithm