Abstract
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,ldots ,ell to some edges and give separate budgets k_{1},k_{2},ldots ,k_{ell } for colors 1,2,ldots ,ell . For every color iin {1,ldots ,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 (i.e., be a cut, a skew multicut, or a directed feedback arc set, respectively), but also needs to satisfy that |Ccap E_{i}|le k_{i} for every iin {1,ldots ,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}+cdots +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 Chainell -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.
Highlights
Graph separation problems are important topics in both theoretical area and applications
The famous minimum cut problem is known to be polynomial-time solvable, many well-known variants are NP-hard, which are intensively studied from the point of view of approximation [1,2,11,13,14,18] and, what is more relevant for this work, parameterized complexity
6 NP-Hardness of MULTI-BUDGETED CUT. It is well-known that the minimum cut problem is polynomial-time solvable, we prove that the Multi- budgeted cut problem is NP-hard for ≥ 2
Summary
Graph separation problems are important topics in both theoretical area and applications. Consider a similar problem Weighted st- cut: given a directed graph G with positive edge weights and two distinguished vertices s, t ∈ V (G), an integer k, and a target weight w, decide if G admits an st-cut of cardinality at most k and weight at most w. 6 we note that Multibudgeted Cut becomes NP-hard for ≥ 2.2 We show that Multi- budgeted Cut is FPT when parameterized by k = k1 + · · · + k For this problem, our branching strategy is as follows. 4. Bound on the number of pushed solutions While we are not able to establish fixedparameter tractability of the weighted variant of the minimum cut problem (even in acyclic graphs) nor of Chain - SAT, we show the following graph-theoretic statement. Our proof is purely existential, and does not yield an enumeration procedure of the “closest to t” solutions
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