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17
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.
Compression via Matroids: A Randomized Polynomial Kernel for Odd Cycle Transversal
"... The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite by deleting at most k of its vertices. In a breakthrough result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a O(4 k kmn) time algorithm for it, the first algorithm with polynomial runtime of ..."
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Cited by 19 (4 self)
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The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite by deleting at most k of its vertices. In a breakthrough result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a O(4 k kmn) time algorithm for it, the first algorithm with polynomial runtime of uniform degree for every fixed k. It is known that this implies a polynomialtime compression algorithm that turns OCT instances into equivalent instances of size at most O(4 k), a socalled kernelization. Since then the existence of a polynomial kernel for OCT, i.e., a kernelization with size bounded polynomially in k, has turned into one of the main open questions in the study of kernelization. Despite the impressive progress in the area, including the recent development of lower bound techniques (Bodlaender
Finding small separators in linear time via treewidth reduction
"... We present a method for reducing the treewidth of a graph while preserving all of its minimal s−t separators up to a certain fixed size k. This technique allows us to solve s−t Cut and Multicut problems with various additional restrictions (e.g., the vertices being removed from the graph form an ind ..."
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Cited by 17 (1 self)
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We present a method for reducing the treewidth of a graph while preserving all of its minimal s−t separators up to a certain fixed size k. This technique allows us to solve s−t Cut and Multicut problems with various additional restrictions (e.g., the vertices being removed from the graph form an independent set or induce a connected graph) in linear time for every fixed number k of removed vertices. Our results have applications for problems that are not directly defined by separators, but the known solution methods depend on some variant of separation. For example, we can solve similarly restricted generalizations of Bipartization (delete at most k vertices from G to make it bipartite) in almost linear time for every fixed number k of removed vertices. These results answer a number of open questions in the area of parameterized complexity. Furthermore, our technique turns out to be relevant for (H,C,K)and (H,C,≤K)coloring problems as well, which are cardinality constrained variants of the classical Hcoloring problem. We make progress in the classification of the parameterized complexity of these problems by identifying new cases that can be solved in almost linear time for every fixed cardinality bound.
FixedParameter Tractability of Directed Multiway Cut Parameterized by the Size of the Cutset
"... Given a directed graph G, a set of k terminals and an integer p, the DIRECTED VERTEX MULTIWAY CUT problem asks if there is a set S of at most p (nonterminal) vertices whose removal disconnects each terminal from all other terminals. DIRECTED EDGE MULTIWAY CUT is the analogous problem where S is a se ..."
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Cited by 14 (5 self)
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Given a directed graph G, a set of k terminals and an integer p, the DIRECTED VERTEX MULTIWAY CUT problem asks if there is a set S of at most p (nonterminal) vertices whose removal disconnects each terminal from all other terminals. DIRECTED EDGE MULTIWAY CUT is the analogous problem where S is a set of at most p edges. These two problems indeed are known to be equivalent. A natural generalization of the multiway cut is the multicut problem, in which we want to disconnect only a set of k given pairs instead of all pairs. Marx (Theor. Comp. Sci. 2006) showed that in undirected graphs multiway cut is fixedparameter tractable (FPT) parameterized by p. Marx and Razgon (STOC 2011) showed that undirected multicut is FPT and directed multicut is W[1]hard parameterized by p. We complete the picture here by our main result which is that both DIRECTED VERTEX MULTIWAY CUT and DIRECTED EDGE MULTIWAY CUT can be solved in time 22O(p) nO(1) , i.e., FPT parameterized by size p of the cutset of the solution. This answers an open question raised by Marx (Theor. Comp. Sci. 2006) and Marx and Razgon (STOC 2011). It follows from our result that DIRECTED MULTICUT is FPT for the case of k = 2 terminal pairs, which answers another open problem raised in Marx and Razgon (STOC 2011). 1
Fixedparameter tractability of multicut in directed acyclic graphs
, 2015
"... The Multicut problem, given a graph G, a set of terminal pairs T = {(si, ti)  1 ≤ i ≤ r}, and an integer p, asks whether one can find a cutset consisting of at most p nonterminal vertices that separates all the terminal pairs, i.e., after removing the cutset, ti is not reachable from si for each ..."
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Cited by 9 (0 self)
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The Multicut problem, given a graph G, a set of terminal pairs T = {(si, ti)  1 ≤ i ≤ r}, and an integer p, asks whether one can find a cutset consisting of at most p nonterminal vertices that separates all the terminal pairs, i.e., after removing the cutset, ti is not reachable from si for each 1 ≤ i ≤ r. The fixedparameter tractability of Multicut in undirected graphs, parameterized by the size of the cutset only, has been recently proved by Marx and Razgon [SIAM J. Comput., 43 (2014), pp. 355–388] and, independently, by Bousquet, Daligault, and Thomassé [Proceedings of STOC, ACM, 2011, pp. 459–468], after resisting attacks as a longstanding open problem. In this paper we prove that Multicut is fixedparameter tractable on directed acyclic graphs when parameterized both by the size of the cutset and the number of terminal pairs. We complement this result by showing that this is implausible for parameterization by the size of the cutset only, as this version of the problem remains W [1]hard.
Parameterized Tractability of Multiway Cut with Parity Constraints
"... Abstract. In this paper, we study a parity based generalization of the classical MULTIWAY CUT problem. Formally, we study the PARITY MULTIWAY CUT problem, where the input is a graph G, vertex subsets Te and To (T = Te ∪ To) called terminals, a positive integer k and the objective is to test whether ..."
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Abstract. In this paper, we study a parity based generalization of the classical MULTIWAY CUT problem. Formally, we study the PARITY MULTIWAY CUT problem, where the input is a graph G, vertex subsets Te and To (T = Te ∪ To) called terminals, a positive integer k and the objective is to test whether there exists a ksized vertex subset S such that S intersects all odd paths from v ∈ To to T \ {v} and all even paths from v ∈ Te to T \ {v}. When Te = To, this is precisely the classical MULTIWAY CUT problem. If To = ∅ then this is the EVEN MULTIWAY CUT problem and if Te = ∅ then this is the ODD MULTIWAY CUT problem. We remark that even the problem of deciding whether there is a set of at most k vertices that intersects all odd paths between a pair of vertices s and t is NPcomplete. Our primary motivation for studying this problem is the recently initiated parameterized study of parity versions of graphs minors (Kawarabayashi, Reed and Wollan, FOCS 2011) and separation problems similar to MULTIWAY CUT. The area of design of parameterized algorithms for graph separation problems has seen a lot of recent activity, which includes algorithms
Completely inapproximable monotone and antimonotone parameterized problems
"... We prove that weighted monotone/antimonotone circuit satisfiability has no fixedparameter tractable approximation algorithm with any approximation ratio function ρ, unless FPT 6 = W [1]. In particular, not having such an fptapproximation algorithm implies that these problems have no polynomialti ..."
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Cited by 6 (0 self)
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We prove that weighted monotone/antimonotone circuit satisfiability has no fixedparameter tractable approximation algorithm with any approximation ratio function ρ, unless FPT 6 = W [1]. In particular, not having such an fptapproximation algorithm implies that these problems have no polynomialtime approximation algorithms with ratio ρ(OPT) for any nontrivial function ρ.
Confronting intractability via parameters
, 2011
"... One approach to confronting computational hardness is to try to understand the contribution of various parameters to the running time of algorithms and the complexity of computational tasks. Almost no computational tasks in real life are specified by their size alone. It is not hard to imagine that ..."
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Cited by 3 (0 self)
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One approach to confronting computational hardness is to try to understand the contribution of various parameters to the running time of algorithms and the complexity of computational tasks. Almost no computational tasks in real life are specified by their size alone. It is not hard to imagine that some parameters contribute more intractability than others and it seems reasonable to develop a theory of computational complexity which seeks to exploit this fact. Such a theory should be able to address the needs of practicioners in algorithmics. The last twenty years have seen the development of such a theory. This theory has a large number of successes in terms of a rich collection of algorithmic techniques both practical and theoretical, and a finegrained intractability theory. Whilst the theory has been widely used in a number of areas of applications including computational biology, linguistics, VLSI design, learning theory and many others, knowledge of the area is highly varied. We hope that this article will show both the basic theory and point at the wide array of techniques available. Naturally the treatment is condensed, and the reader who wants more should go to the texts of Downey and Fellows [125], Flum and Grohe [155], Niedermeier [240], and the upcoming undergraduate text Downey and Fellows [127].
Partitioning into Colorful Components by Minimum Edge Deletions
"... The NPhard Colorful Components problem is, given a vertexcolored graph, to delete a minimum number of edges such that no connected component contains two vertices of the same color. It has applications in multiple sequence alignment and in multiple network alignment where the colors correspond to ..."
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The NPhard Colorful Components problem is, given a vertexcolored graph, to delete a minimum number of edges such that no connected component contains two vertices of the same color. It has applications in multiple sequence alignment and in multiple network alignment where the colors correspond to species. We initiate a systematic complexitytheoretic study of Colorful Components by presenting NPhardness as well as fixedparameter tractability results for different variants of Colorful Components. We also perform experiments with our algorithms and additionally develop an efficient and very accurate heuristic algorithm clearly outperforming a previous mincutbased heuristic on multiple sequence alignment data.
Important separators and parameterized algorithms
, 2011
"... The notion of “important separators” and bounding the number of such separators turned out to be a very useful technique in the design of fixedparameter tractable algorithms for multi(way) cut problems. For example, the recent breakthrough result of Chen et al. [3] on the Directed Feedback Vertex ..."
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The notion of “important separators” and bounding the number of such separators turned out to be a very useful technique in the design of fixedparameter tractable algorithms for multi(way) cut problems. For example, the recent breakthrough result of Chen et al. [3] on the Directed Feedback Vertex Set problem can be also explained using this notion. In my talk, I will overview combinatorial and algorithmic results that can be obtained by studying such separators.