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Fixedparameter tractability of multicut parameterized by the size of the cutset
, 2011
"... Given an undirected graph G, a collection {(s1, t1),...,(sk, tk)} of pairs of vertices, and an integer p, the EDGE MULTICUT problem ask if there is a set S of at most p edges such that the removal of S disconnects every si from the corresponding ti. VERTEX MULTICUT is the analogous problem where S i ..."
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Cited by 32 (6 self)
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Given an undirected graph G, a collection {(s1, t1),...,(sk, tk)} of pairs of vertices, and an integer p, the EDGE MULTICUT problem ask if there is a set S of at most p edges such that the removal of S disconnects every si from the corresponding ti. VERTEX MULTICUT is the analogous problem where S is a set of at most p vertices. Our main result is that both problems can be solved in time 2O(p3) · nO(1), i.e., fixedparameter tractable parameterized by the size p of the cutset in the solution. By contrast, it is unlikely that an algorithm with running time of the form f (p) · nO(1) exists for the directed version of the problem, as we show it to be W[1]hard parameterized by the size of the cutset.
Structural Decomposition Methods and What They are Good For
"... This paper reviews structural problem decomposition methods, such as tree and path decompositions. It is argued that these notions can be applied in two distinct ways: Either to show that a problem is efficiently solvable when a width parameter is fixed, or to prove that the unrestricted (or some wi ..."
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Cited by 3 (1 self)
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This paper reviews structural problem decomposition methods, such as tree and path decompositions. It is argued that these notions can be applied in two distinct ways: Either to show that a problem is efficiently solvable when a width parameter is fixed, or to prove that the unrestricted (or some widthparameter free) version of a problem is tractable by using a widthnotion as a mathematical tool for directly solving the problem at hand. Examples are given for both cases. As a new showcase for the latter usage, we report some recent results on the Partner Units Problem, a form of configuration problem arising in an industrial context. We use the notion of a path decomposition to identify and solve a tractable class of instances of this problem with practical relevance.
MultiMultiway Cut Problem on Graphs of Bounded Branch Width
"... Abstract. The MultiMultiway Cut problem proposed by Avidor and Langberg[2] is a natural generalization of Multicut and Multiway Cut problems. That is, given a simple graphG and c sets of vertices S1, · · · , Sc, the problem asks for a minimum set of edges whose removal disconnects every pair of ..."
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Abstract. The MultiMultiway Cut problem proposed by Avidor and Langberg[2] is a natural generalization of Multicut and Multiway Cut problems. That is, given a simple graphG and c sets of vertices S1, · · · , Sc, the problem asks for a minimum set of edges whose removal disconnects every pair of vertices in Si for all 1 ≤ i ≤ c. In [13], the authors asked whether the problem is polynomial time solvable for fixed c on trees. In this paper, we give both a logical approach and a dynamic programming approach to the MultiMultiway Cut problem on graphs of bounded branch width, which is exactly the class of graphs with bounded treewidth. In fact, for fixed c and branch width k, we show that the MultiMultiway Cut problem can be solved in linear time, thus affirmatively answer the question in [13]. 1