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).

翻译：受软件定义网络中的控制器放置问题与经典“蛋糕切割”公平划分原则的启发，我们研究了以下两人零和博弈。在该模型中，防御者将有限数量的控制器放置在图的顶点上，而攻击者则删除有限数量的顶点。防御者的得分为从任一剩余控制器可达的存活顶点总数。我们形式化了与不同博弈动态（防御者先手、攻击者先手、玩家同时行动、纯策略或混合策略）相关的计算问题。研究表明，这些自然问题根据具体变体属于$\mathsf{NP}$完全或$Σ^\mathsf{P}_2$完全。这些困难性结果为不同解质量定义下的最优控制器放置算法提供了局限性。最后，我们提出了结构洞见，从而得到了针对受限图类（即区间图和有界树宽图）的有效算法。