Multibudgeted Directed Cuts
Kratsch, S and Li, S and Marx, Dániel and Pilipczuk, Marcin and Wahlstrom, Magnus (2020) Multibudgeted Directed Cuts. ALGORITHMICA, 82 (8). pp. 21352155. ISSN 01784617 10.1007/s00453019006091

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Abstract
In this paper, we study multibudgeted 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 Ei be the set of edges of color i. The solution C for the multibudgeted 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 Ei vertical bar <= k(i) for every i is an element of {1,..., l}. Contrary to the classic minimum cut problem, the multibudgeted variant turns out to be NPhard 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 multibudgeted 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 sourcetosink 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 flowguided branching and show an FPT bound on the number of (appropriately defined) important multibudgeted separators. This allows us to extend our algorithm to the Skew Multicut and Directed Feedback Arc Set problems. Furthermore, we show connections of the multibudgeted variants with weighted variants of the directed cut problems and the Chain lSATproblem, whose parameterized complexity remains an open problem. We show that these problems admit a boundedinparameter 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; Multibudgeted 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 
URI:  https://eprints.sztaki.hu/id/eprint/9987 
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