Discounted Cuts: A Stackelberg Approach to Network Disruption

TL;DR

This work studies a Stackelberg variant of cut problems in which an attacker can delete up to $k$ edges to hinder s--t flow, and a defender subsequently reroutes. It formalizes discounted cut costs via $Cost^{exp}_{k}$ and $Cost^{cheap}_{k}$, analyzing eight discounted-cut variants across general and bounded-genus graphs. The main contributions include NP-hardness and W[1]-hardness results on general graphs, and polynomial-time solvability on graphs of bounded genus with polynomial-cost edges, achieved via a genus-aware generating-function framework; planar-case algorithms exploit duality and odd-crossing walks. The results provide a unified framework connecting algorithmic game theory, network interdiction, and operations research, and identify open questions about FPT behavior and planarity with arbitrary weights. The findings enable tractable solutions for structured networks (e.g., transportation/infrastructure) and lay groundwork for future cross-disciplinary collaborations.

Abstract

We study a Stackelberg variant of the classical Most Vital Links problem, modeled as a one-round adversarial game between an attacker and a defender. The attacker strategically removes up to $k$ edges from a flow network to maximally disrupt flow between a source $s$ and a sink $t$, after which the defender optimally reroutes the remaining flow. To capture this attacker--defender interaction, we introduce a new mathematical model of discounted cuts, in which the cost of a cut is evaluated by excluding its $k$ most expensive edges. This model generalizes the Most Vital Links problem and uncovers novel algorithmic and complexity-theoretic properties. We develop a unified algorithmic framework for analyzing various forms of discounted cut problems, including minimizing or maximizing the cost of a cut under discount mechanisms that exclude either the $k$ most expensive or the $k$ cheapest edges. While most variants are NP-complete on general graphs, our main result establishes polynomial-time solvability for all discounted cut problems in our framework when the input is restricted to bounded-genus graphs, a relevant class that includes many real-world networks such as transportation and infrastructure networks. With this work, we aim to open collaborative bridges between artificial intelligence, algorithmic game theory, and operations research.

Figures (1)

• Figure 1: A minimum $s$-$t$-cut in this example is $\{sa, sc\}$ (blue) of cost $6$. For $k=1$, the minimum discounted $s$-$t$-cut is $\{ab, ad, cd\}$ (red), with discounted cost $2$. This corresponds to the edge $cd$ being the most vital link, i.e., the removal of $cd$ leads to the largest decrease in maximum flow.

Theorems & Definitions (31)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 1
• proof : Proof of \ref{['lem:general']}
• Proposition 1
• Proposition 2
• Proposition 3