Multi-Budgeted Directed Cuts

Authors Stefan Kratsch, Shaohua Li, Dániel Marx, Marcin Pilipczuk, Magnus Wahlström

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Author Details

Stefan Kratsch
  • Institut für Informatik, Humboldt-Universität zu Berlin
Shaohua Li
  • Institute of Informatics, University of Warsaw
Dániel Marx
  • Institute for Computer Science and Control, Hungarian Academy of Sciences (MTA SZTAKI)
Marcin Pilipczuk
  • Institute of Informatics, University of Warsaw
Magnus Wahlström
  • Royal Holloway, University of London

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Stefan Kratsch, Shaohua Li, Dániel Marx, Marcin Pilipczuk, and Magnus Wahlström. Multi-Budgeted Directed Cuts. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


In this paper, we study multi-budgeted variants of the classic minimum cut problem and graph separation problems that turned out to be important in parameterized complexity: Skew Multicut and Directed Feedback Arc Set. In our generalization, we assign colors 1,2,...,l to some edges and give separate budgets k_1,k_2,...,k_l for colors 1,2,...,l. For every color i in {1,...,l}, let E_i be the set of edges of color i. The solution C for the multi-budgeted variant of a graph separation problem not only needs to satisfy the usual separation requirements (i.e., be a cut, a skew multicut, or a directed feedback arc set, respectively), but also needs to satisfy that |C cap E_i| <= k_i for every i in {1,...,l}. Contrary to the classic minimum cut problem, the multi-budgeted variant turns out to be NP-hard even for l = 2. We propose FPT algorithms parameterized by k=k_1 +...+ k_l for all three problems. To this end, we develop a branching procedure for the multi-budgeted minimum cut problem that measures the progress of the algorithm not by reducing k as usual, by but elevating the capacity of some edges and thus increasing the size of maximum source-to-sink flow. Using the fact that a similar strategy is used to enumerate all important separators of a given size, we merge this process with the flow-guided branching and show an FPT bound on the number of (appropriately defined) important multi-budgeted separators. This allows us to extend our algorithm to the Skew Multicut and Directed Feedback Arc Set problems. Furthermore, we show connections of the multi-budgeted variants with weighted variants of the directed cut problems and the Chain l-SAT problem, whose parameterized complexity remains an open problem. We show that these problems admit a bounded-in-parameter number of "maximally pushed" solutions (in a similar spirit as important separators are maximally pushed), giving somewhat weak evidence towards their tractability.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • important separators
  • multi-budgeted cuts
  • Directed Feedback Vertex Set
  • fixed-parameter tractability
  • minimum cut


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