Parameterized Complexity Dichotomy for Steiner Multicut

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. This problem generalizes several well-studied graph cut problems, in particular the Multicut problem, which corresponds to the case p = 2. The Multicut problem was recently shown to be fixed-parameter tractable for the parameter k [Marx and Razgon, Bousquet et al., STOC 2011]. The question whether this result generalizes to Steiner Multicut motivates the present work. We answer the question that motivated this work, and in fact provide a dichotomy of the parameterized complexity of Steiner Multicut on general graphs. That is, 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 or that the problem is hard (W[1]-hard or even (para-)NP-complete). Among the many results in the paper, we highlight that: • The edge deletion version of Steiner Multicut is fixed-parameter tractable for the parameter k+ t on general graphs (but has no polynomial kernel, even on trees). The algorithm relies on several new structural lemmas, which decompose the Steiner cut into important separators and minimal s-t cuts, and which only hold for the edge deletion version of the problem. • In contrast, both node deletion versions of Steiner Multicut are W[1]-hard for the parameter k + t on general graphs. • All versions of Steiner Multicut are W[1]-hard for the parameter k, even when p = 3 and the graph is a tree plus one node. This means that the mentioned results of Marx and Razgon, and Bousquet et al. do not generalize to even the most basic instances of Steiner Multicut. Since we allow k, t, p, and tw(G) to be any constants, our characterization includes a dichotomy for Steiner Multicut on trees (for tw(G) = 1) as well as a polynomial time versus NP-hardness dichotomy (by restricting k, t, p, tw(G) to constant or unbounded). ∗Max Planck Institute for Informatics, Saarbrücken, Germany. [email protected]. Karl Bringmann is a recipient of the Google Europe Fellowship in Randomized Algorithms, and this research is supported in part by this Google Fellowship. †Ben Gurion University of the Negev, Israel. [email protected] ‡Cluster of Excellence, Saarbrücken, Germany. [email protected] §Max Planck Institute for Informatics, Saarbrücken, Germany. [email protected] ar X iv :1 40 4. 70 06 v1 [ cs .D S] 2 8 A pr 2 01 4