Complexity of the Multilevel Critical Node Problem

TL;DR

This paper investigates the computational complexity of the Multilevel Critical Node problem, a sequential game involving vaccination and infection strategies on graphs, revealing various NP-complete and higher complexity results.

Contribution

It provides the first complexity analysis of MCN and its subgames, including new NP-complete and higher class results, and develops polynomial algorithms for specific graph classes.

Findings

Identified NP-complete, $oldsymbol{ ext{NP}}$-complete, and higher complexity results for MCN.

Established complexity classifications for various graph types and subgames.

Developed polynomial algorithms for certain graph classes in the protection stage.

Abstract

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, $\Sigma_2^p$-complete and $\Sigma_3^p$-complete problems. Furthermore, for the last level of the game, the protection stage, we build polynomial time algorithms for certain graph classes.