Discounted Cuts: A Stackelberg Approach to Network Disruption

2025-11-13

AI Summary

This paper studies the *most critical edge* problem for network flows under a Stackelberg game framework: an attacker first removes at most $k$ edges to minimize the $s$-$t$ maximum flow, and a defender then optimally reroutes the residual flow. To capture the adversarial interaction, we introduce the *discounted cut* model—where the cost of a cut is defined as the sum of capacities of all its edges except the $k$ most expensive ones. Theoretically, we prove the problem is NP-complete on general graphs but admits a polynomial-time algorithm on bounded-genus graphs (e.g., urban transportation or infrastructure networks). Our work generalizes the classical most critical link problem and bridges AI-driven game theory, combinatorial optimization, and operations research. It provides the first scalable theoretical model and efficient algorithms for robustness analysis of real-world networks.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Planning, Routing, and Scheduling: Planning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Modeling network disruption as a Stackelberg game between attacker and defender
• Introducing discounted cuts to evaluate network vulnerability after edge removal
• Developing algorithms for discounted cut problems on bounded-genus graphs

Innovation

• Discounted cuts model excludes k most expensive edges
• Unified algorithmic framework for various discount mechanisms
• Polynomial-time solvability for bounded-genus graphs