Directed Multicut with linearly ordered terminals

Motivated by an application in network security, we investigate the following “linear” case of Directed Multicut. Let G be a directed graph which includes some distinguished vertices t1, . . . , tk. What is the size of the smallest edge cut which eliminates all paths from ti to tj for all i < j? We show that this problem is fixed-parameter tractable when parametrized in the cutset size p via an algorithm running in O(4pn) time. 1 Multicut requests as partially ordered sets The problem of finding a smallest edge cut separating vertices in a graph has received much attention over the past 50 years. Directed Multicut, one of the more general forms of this problem, encompasses numerous applications in algorithmic graph theory. Name: Directed Multicut. Instance: A directed graph G and pairs of terminal vertices {(s1, t1), . . . , (sk, tk)} from G. Problem: Find a smallest set of edges in G whose deletion eliminates all paths si → ti. Special cases of the Directed Multicut problem have been met with success, although the general problem has no polynomial-time solution unless P = NP. The