Multi-budgeted Directed Cuts

Kratsch, S and Li, S and Marx, Dániel and Pilipczuk, Marcin and Wahlstrom, Magnus (2020) Multi-budgeted Directed Cuts. ALGORITHMICA, 82 (8). pp. 2135-2155. ISSN 0178-4617 10.1007/s00453-019-00609-1


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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 is an element of {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 vertical bar C boolean AND E-i vertical bar <= k(i) for every i is an element of {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-SATproblem, 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.

Item Type: Article
Uncontrolled Keywords: minimum cut; Fixed parameter tractability; Important separators; Multi-budgeted cuts; Directed feedback vertex set;
Subjects: Q Science > QA Mathematics and Computer Science > QA75 Electronic computers. Computer science / számítástechnika, számítógéptudomány
Divisions: Informatics Laboratory
SWORD Depositor: MTMT Injector
Depositing User: MTMT Injector
Date Deposited: 05 Dec 2020 08:58
Last Modified: 17 Nov 2021 13:52

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