Modelling Network Resilience: The Complexity of Some Graph Division Games

TL;DR

This paper models a two-player game involving controller placement and vertex deletion on graphs, analyzing its computational complexity and providing efficient algorithms for specific graph classes.

Contribution

It formalizes the game-theoretic model for network resilience, proving NP- and Sigma2P-completeness, and offers efficient algorithms for interval graphs and bounded treewidth graphs.

Findings

Problems are NP-complete or Sigma2P-complete depending on variants.

Structural insights enable efficient algorithms for certain graph classes.

Complexity results limit the feasibility of optimal controller placement.

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 $\Sigma^\mathsf{P}_2$-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).