Results 1  10
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14
On problems without polynomial kernels
 LECT. NOTES COMPUT. SCI
, 2007
"... Kernelization is a strong and widelyapplied technique in parameterized complexity. In a nutshell, a kernelization algorithm, or simply a kernel, is a polynomialtime transformation that transforms any given parameterized instance to an equivalent instance of the same problem, with size and parame ..."
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Cited by 144 (16 self)
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Kernelization is a strong and widelyapplied technique in parameterized complexity. In a nutshell, a kernelization algorithm, or simply a kernel, is a polynomialtime transformation that transforms any given parameterized instance to an equivalent instance of the same problem, with size and parameter bounded by a function of the parameter in the input. A kernel is polynomial if the size and parameter of the output are polynomiallybounded by the parameter of the input. In this paper we develop a framework which allows showing that a wide range of FPT problems do not have polynomial kernels. Our evidence relies on hypothesis made in the classical world (i.e. nonparametric complexity), and evolves around a new type of algorithm for classical decision problems, called a distillation algorithm, which might be of independent interest. Using the notion of distillation algorithms, we develop a generic lowerbound engine which allows us to show that a variety of FPT problems, fulfilling certain criteria, cannot have polynomial kernels unless the polynomial hierarchy collapses. These problems include kPath, kCycle, kExact Cycle, kShort Cheap Tour, kGraph Minor Order Test, kCutwidth, kSearch Number, kPathwidth, kTreewidth, kBranchwidth, and several optimization problems parameterized by treewidth or cliquewidth.
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 33 (5 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.
Constant ratio fixedparameter approximation of the edge multicut problem
 In ESA 2009
, 2009
"... Abstract. The input of the Edge Multicut problem consists of an undirected graph G and pairs of terminals {s1, t1},..., {sm, tm}; the task is to remove a minimum set of edges such that si and ti are disconnected for every 1 ≤ i ≤ m. The parameterized complexity of the problem, parameterized by the ..."
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Cited by 18 (3 self)
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Abstract. The input of the Edge Multicut problem consists of an undirected graph G and pairs of terminals {s1, t1},..., {sm, tm}; the task is to remove a minimum set of edges such that si and ti are disconnected for every 1 ≤ i ≤ m. The parameterized complexity of the problem, parameterized by the maximum number k of edges that are allowed to be removed, is currently open. The main result of the paper is a parameterized 2approximation algorithm: in time f(k) · nO(1), we can either find a solution of size 2k or correctly conclude that there is no solution of size k. The proposed algorithm is based on a transformation of the Edge Multicut problem into a variant of parameterized Max2SAT problem, where the parameter is related to the number of clauses that are not satisfied. It follows from previous results that the latter problem can be 2approximated in a fixedparameter time; on the other hand, we show here that it is W[1]hard. Thus the additional contribution of the present paper is introducing the first natural W[1]hard problem that is constantratio fixedparameter approximable. 1
Multicut is FPT
 In STOC
, 2011
"... Let G = (V,E) be a graph on n vertices and R be a set of pairs of vertices in V called requests. A multicut is a subset F of E such that every request xy of R is separator by F, i.e.every xypath of G intersects F. We show that there exists an O(f(k)nc) algorithm which decides if there exists a mult ..."
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Cited by 17 (0 self)
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Let G = (V,E) be a graph on n vertices and R be a set of pairs of vertices in V called requests. A multicut is a subset F of E such that every request xy of R is separator by F, i.e.every xypath of G intersects F. We show that there exists an O(f(k)nc) algorithm which decides if there exists a multicut of size at most k. In other words, the MULTICUT problem parameterized by the solution size k is FixedParameter Tractable. 1
Exact algorithms and applications for Treelike Weighted Set Cover
 JOURNAL OF DISCRETE ALGORITHMS
, 2006
"... We introduce an NPcomplete special case of the Weighted Set Cover problem and show its fixedparameter tractability with respect to the maximum subset size, a parameter that appears to be small in relevant applications. More precisely, in this practically relevant variant we require that the given ..."
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Cited by 10 (5 self)
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We introduce an NPcomplete special case of the Weighted Set Cover problem and show its fixedparameter tractability with respect to the maximum subset size, a parameter that appears to be small in relevant applications. More precisely, in this practically relevant variant we require that the given collection C of subsets of a some base set S should be “treelike.” That is, the subsets in C can be organized in a tree T such that every subset onetoone corresponds to a tree node and, for each element s of S, the nodes corresponding to the subsets containing s induce a subtree of T. This is equivalent to the problem of finding a minimum edge cover in an edgeweighted acyclic hypergraph. Our main result is an algorithm running in O(3 k ·mn) time where k denotes the maximum subset size, n: = S, and m: = C. The algorithm also implies a fixedparameter tractability result for the NPcomplete Multicut in Trees problem, complementing previous approximation results. Our results find applications in computational biology in phylogenomics and for saving memory in tree decomposition based graph algorithms.
Complexity and exact algorithms for multicut
 In: SOFSEM
"... Abstract. The Multicut problem is defined as: given an undirected graph and a collection of pairs of terminal vertices, find a minimum set of edges or vertices whose removal disconnects each pair. We mainly focus on the case of removing vertices, where we distinguish between allowing or disallowing ..."
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Cited by 8 (0 self)
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Abstract. The Multicut problem is defined as: given an undirected graph and a collection of pairs of terminal vertices, find a minimum set of edges or vertices whose removal disconnects each pair. We mainly focus on the case of removing vertices, where we distinguish between allowing or disallowing the removal of terminal vertices. Complementing and refining previous results from the literature, we provide several NPcompleteness and (fixedparameter) tractability results for restricted classes of graphs such as trees, interval graphs, and graphs of bounded treewidth. 1
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.
Treewidth reduction for the parameterized Multicut problem
, 2010
"... The parameterized Multicut problem consists in deciding, given a graph, a set of requests (i.e. pairs of vertices) and an integer k, whether there exists a set of k edges which disconnects the two endpoints of each request. Determining whether Multicut is FixedParameter Tractable with respect to k ..."
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Cited by 1 (1 self)
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The parameterized Multicut problem consists in deciding, given a graph, a set of requests (i.e. pairs of vertices) and an integer k, whether there exists a set of k edges which disconnects the two endpoints of each request. Determining whether Multicut is FixedParameter Tractable with respect to k is one of the most important open question in parameterized complexity [5]. We show that Multicut reduces to instances of treewidth bounded in k. To that aim, we establish new reduction rules that apply to arbitrary instances of Multicut. Based on graph separability properties, these rules identify an irrelevant request that can be safely removed. As a main consequence, these rules imply that the degree of the request graph of any instance is bounded by a function of k. We prove that when the input graph has a large clique minor or a large grid minor, then we can remove an irrelevant request or contract an edge.
Multicut viewed through the eyes of vertex cover
"... We take a new look at the multicut problem in trees through the eyes of the vertex cover problem. This connection, together with other techniques that we develop, allows us to significantly improve the O(k 6) upper bound on the kernel size for multicut, given by Bousquet et al., to O(k 3). We exploi ..."
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We take a new look at the multicut problem in trees through the eyes of the vertex cover problem. This connection, together with other techniques that we develop, allows us to significantly improve the O(k 6) upper bound on the kernel size for multicut, given by Bousquet et al., to O(k 3). We exploit this connection further to present a parameterized algorithm for multicut that runs in time O ∗ (ρ k), where ρ = ( √ 5 + 1)/2 ≈ 1.618. This improves the previous (time) upper bound of O ∗ (2 k), given by Guo and Niedermeier, for the problem. 1
Introduction to Complexity Theory Column 54
"... about small depth quantum circuits and Salil Vadhan writing on the complexity of zero knowledge. And warmest thanks to Jiong Guo and Rolf Niedermeier for this issue’s article on data reduction ..."
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about small depth quantum circuits and Salil Vadhan writing on the complexity of zero knowledge. And warmest thanks to Jiong Guo and Rolf Niedermeier for this issue’s article on data reduction