Abstract
We consider the Steiner Multicut problem, which asks, given an undirected graph G, a collection T={T1,…,Tt}, Ti⊆V(G), of terminal sets of size at most p, and an integer k, whether there is a set S of at most k edges or nodes such that of each set Ti at least one pair of terminals is in different connected components of G−S. We provide a dichotomy of the parameterized complexity of Steiner Multicut. For any combination of k, t, p, and the treewidth tw(G) as constant, parameter, or unbounded, and for all versions of the problem (edge deletion and node deletion with and without deletable terminals), we prove either that the problem is fixed-parameter tractable, W[1]-hard, or (para-)NP-complete. Our characterization includes a dichotomy for Steiner Multicut on trees as well as a polynomial time versus NP-hardness dichotomy (by restricting k,t,p,tw(G) to constant or unbounded).
Highlights
Graph cut problems are among the most fundamental problems in algorithmic research
The classic result in this area is the polynomial-time algorithm for the s–t cut problem of Ford and Fulkerson [19]. This result inspired a research program to discover the computational complexity of this problem and of more general graph cut problems
In a recent major advance of the research program on graph cut problems, Bousquet et al [3] and Marx and Razgon [32] showed that Multicut is fixed-parameter tractable in the size k of the cut only, meaning that it has an algorithm running in time f (k) · poly(|V (G)|) for some function f, resolving a longstanding problem in parameterized complexity
Summary
The classic result in this area is the polynomial-time algorithm for the s–t cut problem of Ford and Fulkerson [19] (independently proven by Elias et al [17] and Dantzig and Fulkerson [13]) This result inspired a research program to discover the computational complexity of this problem and of more general graph cut problems. We continue the research program on generalized graph cut problems, and consider the Steiner Multicut problem. This problem was proposed by Klein et al [25], and appears in several versions, depending on whether we want to delete edges or nodes, and whether we are allowed to delete terminal nodes. To the best of our knowledge, Steiner Multicut in its general form has not yet been considered from the perspective of parameterized complexity
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