Cardinality-Constrained Critical Node Detection Problem

We consider methodologies for managing risk in a telecommunication network based on identification of the critical nodes. The objective is to minimize the number of vertices whose deletion results in disconnected components which are constrained by a given cardinality. This is referred to as the CARDINALITY CONSTRAINED CRITICAL NODE PROBLEM (CC-CNP), and finds application in epidemic control, telecommunications, and military tactical planning, among others. From a telecommunication perspective, the set of critical nodes helps determine which players should be removed from the network in the event of a virus outbreak. Conversely, in order to maintain maximum global connectivity, it should be ensured that the critical nodes remain intact and as secure as possible. The presence of these nodes make a network vulnerable to attacks as they are crucial for the overall connectivity. This is a variation of the CRITICAL NODE DETECTION PROBLEM which has a known complexity and heuristic procedure. In this chapter, we review the recent work in this area, provide formulations based on integer linear programming and develop heuristic procedures for CC-CNP. We also examine the relations of CC-CNP with the well known NODE DELETION PROBLEM and discuss complexity results as a result of this relation.