The Complexity of the Infinity Replacement Problem in the Cyber Security Model

Abstract

In this article we present a new defensive problem on a security system. Let M be a security model (T, C,P), where T= (V, E) is a rooted tree rooted at r,C is a multiset of E(T) costs and P is a multiset of V(T)-1 prizes and let (T, c,p) be a security system, where c and p are bijections c: E(T) C and p: V(T) r P, respectively. Given a budget BZ+, we consider the problem of determining the existence of an edge e E(T), where c(e)=i such that the maximum total of prizes obtained from an optimal attack in the security system (T, c,p) is minimum when i is replaced by ∞. We define the decision and optimization versions of the problem and examine their computational complexities. We prove that the decision problem is NP-hard and provide a pseudopolynomial time algorithm for the optimization version of the problem. Additionally, we show that some restricted instances can be solved in polynomial time. In the end conclusion and an example of the application of the algorithm are given.