多层临界点问题的复杂性
Complexity of the Multilevel Critical Node Problem
摘要：
在这项工作中，我们分析了一个连续的游戏在一个图称为多级关键节点问题（MCN）。一个防守者和一个进攻者是这个博弈的参与者。防御者从预防性地阻止顶点（接种疫苗）被攻击开始。然后，攻击者感染未接种疫苗的顶点的子集，最后，防御者用保护策略做出反应。我们提供的第一个计算复杂性的结果与MCN及其子游戏。此外，通过考虑酉，加权，无向和有向图，我们澄清了这些问题的理论易处理性如何变化。我们的研究结果有助于新的NP-完全，$Σ_2^p$ -完全和$Σ_3^p$ -完全问题。此外，对于游戏的最后一级，保护阶段，我们建立了多项式时间算法的某些图形类。
英文摘要：
In this work, we analyze a sequential game played in a graph called the Multilevel Critical Node problem (MCN). A defender and an attacker are the players of this game. The defender starts by preventively interdicting vertices (vaccination) from being attacked. Then, the attacker infects a subset of non-vaccinated vertices and, finally, the defender reacts with a protection strategy. We provide the first computational complexity results associated with MCN and its subgames. Moreover, by considering unitary, weighted, undirected, and directed graphs, we clarify how the theoretical tractability of those problems vary. Our findings contribute with new NP-complete, $Σ_2^p$-complete and $Σ_3^p$-complete problems. Furthermore, for the last level of the game, the protection stage, we build polynomial time algorithms for certain graph classes.
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