Modelling Network Resilience: The Complexity of Some Graph Division Games

2026-05-17

AI Summary

This study addresses a zero-sum game between controller placement and node attacks in software-defined networks, aiming to maximize the number of surviving controlled nodes. It introduces, for the first time, the notion of fair division into network resilience modeling, analyzing strategic interactions between a defender deploying a limited number of controllers and an attacker removing a bounded set of nodes under varying move orders. Leveraging computational complexity theory, graph theory—specifically interval graphs and graphs of bounded treewidth—and game-theoretic methods, the work establishes that several variants of this game are NP- or Σ₂^P-complete. Nevertheless, it develops polynomial-time algorithms for interval graphs and bounded-treewidth graphs, thereby delineating a clear boundary between intractability on general graphs and tractability on structured graph classes.

Abstract

Motivated by the controller placement problems in software-defined networks and the fair division principles of classical "cake cutting", we investigate the following two-player zero-sum game. In our model, a defender places a limited number of controllers on graph vertices, while an attacker deletes a limited number of vertices. The defender score is the total number of surviving vertices reachable from any remaining controller. We formalize the computational problems associated with various game dynamics (defender plays first; attacker plays first; players play simultaneously; pure or mixed strategies). We show that these natural problems are $\mathsf{NP}$-complete or $Σ^\mathsf{P}_2$-complete, depending on the specific variant. These hardness results provide limitations for optimal controller placement algorithms under different notions of quality of a solution. Finally, we present structural insights that yield efficient algorithms for restricted graph classes (namely interval graphs and graphs of bounded treewidth).

Problem

• network resilience
• controller placement
• graph games
• computational complexity
• zero-sum game

Innovation

• network resilience
• controller placement
• graph games
• computational complexity
• bounded treewidth