A Note on Interdiction of Linear Minimization Problems

TL;DR

This paper analyzes a linear minimization interdiction problem, simplifying the FPTAS approach by focusing on dualized weights and approximate minimizers, leading to potential exact algorithms.

Contribution

It isolates a non-cut-based part of the FPTAS for connectivity interdiction, enabling exact solutions via optimization of reweighted problems and enumeration of approximate minimizers.

Findings

Optimal interdiction witness is a strict 2-approximate minimizer at a specific Lagrange multiplier.

Exact algorithms are possible if one can optimize the reweighted problem and enumerate its 2-approximate minimizers.

The approach simplifies the interdiction problem by absorbing deletion into truncated weights.

Abstract

Motivated by the FPTAS for connectivity interdiction of Huang et al. (IPCO'24), we isolate the part of the argument that does not use cuts. The setting is a minimization problem over a feasible-set family $\mathcal F$ with a linear objective $w(S)=\sum_{e\in S}w(e)$. After dualizing the interdiction budget, deletion can be absorbed into truncated weights $w_\lambda(e)=\min\{w(e),\lambda c(e)\}$. At an optimal Lagrange multiplier $\lambda^*$, the unknown optimal interdiction witness is a strict $2$-approximate minimizer of the reweighted problem. Thus an exact algorithm can be obtained whenever one can optimize $w_{\lambda^*}$ over $\mathcal F$, enumerate all its $2$-approximate minimizers, and solve the remaining knapsack problem.