On the Bicriterion Maximum Flow Network Interdiction Problem

TL;DR

This paper studies a biobjective network interdiction problem with two capacities per arc, proving its NP-completeness and proposing algorithms for specific graph classes and approximation schemes.

Contribution

It introduces a biobjective extension of the maximum flow interdiction problem, analyzes its computational complexity, and provides algorithms for series-parallel graphs and approximation schemes.

Findings

The problem is NP-complete.

A pseudopolynomial algorithm exists for series-parallel graphs.

A fully polynomial-time approximation scheme is developed for unit interdiction costs.

Abstract

This article focuses on a biobjective extension of the maximum flow network interdiction problem, where each arc in the network is associated with two capacity values. Two maximum flows from a source to a sink are to be computed independently of each other with respect to the first and second capacity function, respectively, while an interdictor aims to minimize the value of both maximum flows by interdicting arcs. We show that this problem is intractable and that the decision problem, which asks whether or not a feasible interdiction strategy is efficient, is NP-complete. We propose a pseudopolynomial time algorithm in the case of two-terminal series-parallel graphs and positive integer-valued interdiction costs. We extend this algorithm to a fully polynomial-time approximation scheme for the case of unit interdiction costs by appropriately partitioning the objective space.