Abstract
The Multicut problem, given a graph G, a set of terminal pairs \(\ensuremath{\mathcal{T}}=\{(s_i,t_i)\ |\ 1\leq i\leq r\}\) and an integer p, asks whether one can find a cutset consisting of at most p non-terminal vertices that separates all the terminal pairs, i.e., after removing the cutset, t i is not reachable from s i for each 1 ≤ i ≤ r. The fixed-parameter tractability of Multicut in undirected graphs, parameterized by the size of the cutset only, has been recently proven by Marx and Razgon [2] and, independently, by Bousquet et al. [3], after resisting attacks as a long-standing open problem. In this paper we prove that Multicut is fixed-parameter tractable on directed acyclic graphs, when parameterized both by the size of the cutset and the number of terminal pairs. We complement this result by showing that this is implausible for parameterization by the size of the cutset only, as this version of the problem remains W[1]-hard.
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References
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Kratsch, S., Pilipczuk, M., Pilipczuk, M., Wahlström, M. (2012). Fixed-Parameter Tractability of Multicut in Directed Acyclic Graphs. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds) Automata, Languages, and Programming. ICALP 2012. Lecture Notes in Computer Science, vol 7391. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31594-7_49
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DOI: https://doi.org/10.1007/978-3-642-31594-7_49
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