Efficient Algorithms for Interdicting Facilities in Trees and Bounded Treewidth Graphs

2026-05-27

AI Summary

This study addresses the network interdiction coverage problem on trees and bounded-treewidth graphs, where the goal is to maximize the total weight of clients disconnected from all facilities by removing at most $r$ edges (REIC) or facilities (RFIC). The authors propose efficient algorithms based on tree decomposition and fixed-parameter dynamic programming, significantly improving the time complexity for REIC on trees from $O(n^7 r)$ to $O(nr^2)$. They formally define the RFIC problem for the first time, prove its NP-completeness, and present an efficient algorithm for trees. Experimental results demonstrate that the proposed methods substantially outperform existing approaches and the Gurobi solver on random tree networks, exhibiting faster runtimes and insensitivity to instance size and topology. Additionally, the framework yields an $O(n^3)$ algorithm for the SSBVE problem on trees.

Abstract

Given a graph $G$ of $n$ nodes partitioned into facilities and customers, the $r$-edge interdiction covering problem (REIC) is to remove up to $r$ edges so as to maximize the total weight of customers disconnected from all facilities, which is called the covering objective function. While REIC is known to be NP-complete for general graphs, Fröhlich and Ruzika show that the problem can be solved in polynomial time when $G$ is a tree, providing an $O(n^7 r)$-time algorithm. We give an efficient $O(nr^2)$-time dynamic programming algorithm for REIC on trees that is fixed-parameter linear in $n$. Evaluating our solution on a benchmark of randomly generated tree networks with baselines of the Fröhlich and Ruzika algorithm and the Gurobi integer program solver, we demonstrate that in practice, our algorithm is both significantly faster and less sensitive to network topology and size. We extend our algorithm for REIC to graphs of bounded treewidth, a well-studied family of sparse graphs that generalizes trees, and obtain a matching runtime of $O(nr^2)$. We also consider the $r$-facility interdiction covering problem (RFIC), a novel variant of this network interdiction problem where the goal is to remove up to $r$ facilities to maximize the covering objective function over disconnected customers. We show that RFIC is NP-complete by observing it generalizes the small set bipartite vertex expansion problem (SSBVE), also known as the minimum $p$-union problem. We give an $O(nr^2)$-time algorithm for RFIC on trees, which also gives an $O(n^3)$-time algorithm for SSBVE on trees.

Problem

• network interdiction
• covering problem
• tree
• bounded treewidth
• facility location

Innovation

• tree algorithms
• network interdiction
• bounded treewidth
• dynamic programming
• facility interdiction