Results 1  10
of
19
On problems without polynomial kernels
 Lect. Notes Comput. Sci
, 2007
"... Abstract. 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 an ..."
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Cited by 65 (9 self)
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Abstract. 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. 1
Parameterized complexity and approximation algorithms
 Comput. J
, 2006
"... Approximation algorithms and parameterized complexity are usually considered to be two separate ways of dealing with hard algorithmic problems. In this paper, our aim is to investigate how these two fields can be combined to achieve better algorithms than what any of the two theories could offer. We ..."
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Cited by 26 (1 self)
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Approximation algorithms and parameterized complexity are usually considered to be two separate ways of dealing with hard algorithmic problems. In this paper, our aim is to investigate how these two fields can be combined to achieve better algorithms than what any of the two theories could offer. We discuss the different ways parameterized complexity can be extended to approximation algorithms, survey results of this type and propose directions for future research. 1.
Linear problem kernels for NPhard problems on planar graphs
 In Proc. 34th ICALP, volume 4596 of LNCS
, 2007
"... Abstract. We develop a generic framework for deriving linearsize problem kernels for NPhard problems on planar graphs. We demonstrate the usefulness of our framework in several concrete case studies, giving new kernelization results for Connected Vertex Cover, Minimum Edge Dominating Set, Maximum ..."
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Cited by 15 (4 self)
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Abstract. We develop a generic framework for deriving linearsize problem kernels for NPhard problems on planar graphs. We demonstrate the usefulness of our framework in several concrete case studies, giving new kernelization results for Connected Vertex Cover, Minimum Edge Dominating Set, Maximum Triangle Packing, and Efficient Dominating Set on planar graphs. On the route to these results, we present effective, problemspecific data reduction rules that are useful in any approach attacking the computational intractability of these problems. 1
On problems without polynomial kernels (extended abstract
 ICALP (1), volume 5125 of LNCS
, 2008
"... Abstract. Kernelization is a central technique used in parameterized algorithms, and in other approaches for coping with NPhard problems. In this paper, we introduce a new method which allows us to show that many problems do not have polynomial size kernels under reasonable complexitytheoretic ass ..."
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Cited by 15 (1 self)
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Abstract. Kernelization is a central technique used in parameterized algorithms, and in other approaches for coping with NPhard problems. In this paper, we introduce a new method which allows us to show that many problems do not have polynomial size kernels under reasonable complexitytheoretic assumptions. These problems include k
Finding a minimum feedback vertex set in time O(1.7548 n
 in Proceedings of the 2nd International Workshop on Parameterized and Exact Computation (IWPEC 2006
, 2006
"... Abstract. We present an O(1.7548 n) algorithm finding a minimum feedback vertex set in a graph on n vertices. ..."
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Cited by 14 (5 self)
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Abstract. We present an O(1.7548 n) algorithm finding a minimum feedback vertex set in a graph on n vertices.
Short cycles make Whard problems hard: FPT algorithms for Whard problems in graphs with no short cycles
, 2006
"... We show that several problems that are hard for various parameterized complexity classes on general graphs, become fixed parameter tractable on graphs with no small cycles. More specifically, we give fixed parameter tractable algorithms for Dominating Set, tVertex Cover (where we need to cover at ..."
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Cited by 11 (6 self)
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We show that several problems that are hard for various parameterized complexity classes on general graphs, become fixed parameter tractable on graphs with no small cycles. More specifically, we give fixed parameter tractable algorithms for Dominating Set, tVertex Cover (where we need to cover at least t edges) and several of their variants on graphs with girth at least five. These problems are known to be W [i]hard for some i ≥ 1 in general graphs. We also show that the Dominating Set problem is W [2]hard for bipartite graphs and hence for triangle free graphs. In the case of Independent Set and several of its variants, we show these problems to be fixed parameter tractable even in triangle free graphs. In contrast, we show that the Dense Subgraph problem where one is interested in finding an induced subgraph on k vertices having at least l edges, paramaterized by k, is W [1]hard even on graphs with girth at least six. Finally, we give an O(log p) ratio approximation algorithm for the Dominating Set problem for graphs with girth at least 5, where p is the size of an optimum dominating set of the graph. This improves the previous O(log n) factor approximation algorithm for the problem, where n is the number of vertices of the input graph.
Faster Steiner Tree Computation in PolynomialSpace
, 2008
"... Given an nnode graph and a subset of k terminal nodes, the NPhard Steiner tree problem is to compute a minimumsize tree which spans the terminals. All the known algorithms for this problem which improve on trivial O(1.62 n)timeenumerationarebasedondynamic programming, and require exponential sp ..."
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Cited by 5 (1 self)
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Given an nnode graph and a subset of k terminal nodes, the NPhard Steiner tree problem is to compute a minimumsize tree which spans the terminals. All the known algorithms for this problem which improve on trivial O(1.62 n)timeenumerationarebasedondynamic programming, and require exponential space. Motivated by the fact that exponentialspace algorithms are typically impractical, in this paper we address the problem of designing faster polynomialspace algorithms. Our first contribution is a simple polynomialspace O(6 k n O(log k))time algorithm, based on a variant of the classical treeseparator theorem. This improves on trivial O(n k+O(1))enumeration for, roughly, k ≤ n/4. Combining the algorithm above (for small k), with an improved branching strategy (for large k), we obtain an O(1.60 n)time polynomialspace algorithm. The refined branching is based on a charging mechanism which shows that, for large values of k, convenient local configurations of terminals and nonterminals must exist. The analysis of the algorithm relies on the Measure & Conquer approach: the nonstandard measure used here is a linear combination of the number of nodes and number of nonterminals. As a byproduct of our work, we also improve the (exponentialspace) time complexity of the problem from O(1.42 n) to O(1.36 n).
Vertex and edge covers with clustering properties: Complexity and algorithms
 In Algorithms and Complexity in Durham
, 2006
"... We consider the concepts of a ttotal vertex cover and a ttotal edge cover (t ≥ 1), which generalize the notions of a vertex cover and an edge cover, respectively. A ttotal vertex (respectively edge) cover of a connected graph G is a vertex (edge) cover S of G such that each connected component of ..."
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Cited by 5 (1 self)
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We consider the concepts of a ttotal vertex cover and a ttotal edge cover (t ≥ 1), which generalize the notions of a vertex cover and an edge cover, respectively. A ttotal vertex (respectively edge) cover of a connected graph G is a vertex (edge) cover S of G such that each connected component of the subgraph of G induced by S has least t vertices (edges). These definitions are motivated by combining the concepts of clustering and covering in graphs. Moreover they yield a spectrum of parameters that essentially range from a vertex cover to a connected vertex cover (in the vertex case) and from an edge cover to a spanning tree (in the edge case). For various values of t, we present N Pcompleteness and approximability results (both upper and lower bounds) and FPT algorithms for problems concerned with finding the minimum size of a ttotal vertex cover, ttotal edge cover and connected vertex cover, in particular improving on a previous FPT algorithm for the latter problem. 1
Complexity and approximation results for the connected vertex cover problem
"... We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APXhard in bipartite graphs and is 5/3approximable in any class of graphs wher ..."
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Cited by 3 (0 self)
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We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APXhard in bipartite graphs and is 5/3approximable in any class of graphs where the vertex cover problem is polynomial (in particular in bipartite graphs).
Improved Upper Bounds for Partial Vertex Cover
, 2008
"... The Partial Vertex Cover problem is to decide whether a graph contains at most k nodes covering at least t edges. We present deterministic and randomized algorithms with run times of O ∗ (1.396 t) and O ∗ (1.2993 t), respectively. For graphs of maximum degree three, we show how to solve this proble ..."
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Cited by 2 (0 self)
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The Partial Vertex Cover problem is to decide whether a graph contains at most k nodes covering at least t edges. We present deterministic and randomized algorithms with run times of O ∗ (1.396 t) and O ∗ (1.2993 t), respectively. For graphs of maximum degree three, we show how to solve this problem in O ∗ (1.26 t) steps. Finally, we give an O ∗ (3 t) algorithm for Exact Partial Vertex Cover, which asks for at most k nodes covering exactly t edges.