Logical 𝑠-𝑡 Min-Cut Problem: An Extension to the Classic 𝑠-𝑡 Min-Cut Problem

Let G be a weighted digraph, s and t be two vertices of G, and t is reachable from s. The logical 𝑠-𝑡 min-cut (LSTMC) problem states how t can be made unreachable from s by removal of some edges of G where (a) the sum of weights of the removed edges is minimum and (b) all outgoing edges of any vertex of G cannot be removed together. If we ignore the second constraint, called the logical removal, the LSTMC problem is transformed to the classic 𝑠-𝑡 min-cut problem. The logical removal constraint applies in situations where non-logical removal is either infeasible or undesired. Although the 𝑠-𝑡 min-cut problem is solvable in polynomial time by the max- ow min-cut theorem, this paper shows the LSTMC problem is NP-Hard, even if G is a DAG with an out-degree of two. Moreover, this paper shows that the LSTMC problem cannot be approximated within logn in a 𝐷𝐴𝐺 with n vertices for some constant . The application of the LSTMC problem is also presented in test case generation of a computer program.