Directed Reachability-Preserving Minimum Edge Cut: Approximation and Planar Hardness

2026-06-16

AI Summary

This study addresses the minimum edge cut problem in three-terminal directed graphs: disconnecting all paths from a source $s_1$ to a sink $t$ while preserving reachability from $s_1$ to another source $s_2$. By modeling the problem via root-cut functions and leveraging polyhedral approximation together with path-cut structures, the authors present the first $O(\sqrt{r})$-approximation algorithm, where $r$ denotes the number of relevant terminals or constraints. For directed acyclic graphs, they further refine the guarantee to $O(\min\{\sqrt{r}, h\})$ by exploiting path-length information, with $h$ being the longest $s_1$–$t$ path length. The problem is also shown to be NP-hard even on planar directed graphs. Key technical contributions include a rooted linear programming relaxation, a finite bimodular node-cut construction, and a node-to-edge transformation for planar digraphs, yielding an $O(\sqrt{n})$-approximation in general directed graphs.

Abstract

We study a directed version of the three-terminal reachability-preserving minimum edge cut problem. Given a directed graph $G=(V,A)$ with arc costs and terminals $s_1,s_2,t$, the one-way directed RPMEC problem asks for a minimum-cost set of arcs whose deletion preserves the reachability $s_1\leadsto s_2$ while destroying the reachability $s_1\leadsto t$. We first give a path--cut formulation in terms of a rooted directed cut function. Using a root-linear approximation for the associated polymatroid, we obtain an $O(\sqrt r)$-approximation, where $r$ is the number of relevant vertices with positive singleton cut value. In particular this gives an $O(\sqrt n)$-approximation in general directed graphs. For acyclic directed graphs, we give an additional singleton-length algorithm and obtain an $O(\min\{\sqrt r,h\})$ guarantee, where $h$ is the maximum number of relevant vertices on an $s_1$-$s_2$ path. Finally, we prove that directed planar RPMEC is NP-hard, even on acyclic planar digraphs with nonnegative costs, by reducing from independent set on cubic planar graphs through a finite-bimodal directed node-cut construction and a planar node-to-edge split.

Problem

• directed graph
• reachability-preserving
• minimum edge cut
• NP-hardness
• approximation

Innovation

• reachability-preserving
• minimum edge cut
• approximation algorithm
• directed planar graphs
• polymatroid