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- W2949645178 abstract "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 $1,2,...,ell$ to some edges and give separate budgets $k_{1},k_{2},...,k_{ell}$. 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, but also needs to satisfy that $|Ccap E_{i}|leq k_{i}$ for every $iin {1,...,ell}$. Contrary to the classic minimum cut problem, the multi-budgeted variant turns out to be NP-hard even for $ell = 2$. We propose FPT algorithms parameterized by $k=k_{1}+...+k_{ell}$ 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 $ell$-SAT problem, whose parameterized complexity remains an open problem. We show that these problems admit a bounded-in-parameter number of pushed solutions (in a similar spirit as important separators are maximally pushed), giving somewhat weak evidence towards their tractability." @default.
- W2949645178 created "2019-06-27" @default.
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- W2949645178 date "2018-10-16" @default.
- W2949645178 modified "2023-09-23" @default.
- W2949645178 title "Multi-budgeted directed cuts" @default.
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