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323
Reflections on multivariate algorithmics and problem parameterization
 PROC. 27TH STACS
, 2010
"... Research on parameterized algorithmics for NPhard problems has steadily grown over the last years. We survey and discuss how parameterized complexity analysis naturally develops into the field of multivariate algorithmics. Correspondingly, we describe how to perform a systematic investigation and e ..."
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Cited by 24 (19 self)
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Research on parameterized algorithmics for NPhard problems has steadily grown over the last years. We survey and discuss how parameterized complexity analysis naturally develops into the field of multivariate algorithmics. Correspondingly, we describe how to perform a systematic investigation and exploitation of the “parameter space” of computationally hard problems.
FixedParameter Algorithms for Cluster Vertex Deletion
, 2008
"... We initiate the first systematic study of the NPhard Cluster Vertex Deletion (CVD) problem (unweighted and weighted) in terms of fixedparameter algorithmics. In the unweighted case, one searches for a minimum number of vertex deletions to transform a graph into a collection of disjoint cliques. Th ..."
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Cited by 23 (11 self)
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We initiate the first systematic study of the NPhard Cluster Vertex Deletion (CVD) problem (unweighted and weighted) in terms of fixedparameter algorithmics. In the unweighted case, one searches for a minimum number of vertex deletions to transform a graph into a collection of disjoint cliques. The parameter is the number of vertex deletions. We present efficient fixedparameter algorithms for CVD applying the fairly new iterative compression technique. Moreover, we study the variant of CVD where the maximum number of cliques to be generated is prespecified. Here, we exploit connections to fixedparameter algorithms for (weighted) Vertex Cover.
A quadratic kernel for feedback vertex set
 in Proc. 20th SODA, ACM/SIAM, 2009
"... We prove that given an undirected graph G on n vertices and an integer k, one can compute in polynomial time in n a graph G ′ with at most 5k 2 +k vertices and an integer k ′ such that G has a feedback vertex set of size at most k iff G ′ has a feedback vertex set of size at most k ′. This result im ..."
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Cited by 22 (2 self)
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We prove that given an undirected graph G on n vertices and an integer k, one can compute in polynomial time in n a graph G ′ with at most 5k 2 +k vertices and an integer k ′ such that G has a feedback vertex set of size at most k iff G ′ has a feedback vertex set of size at most k ′. This result improves a previous O(k 11) kernel of Burrage et al. [6], and a more recent cubic kernel of Bodlaender [3]. This problem was communicated by Fellows in [5]. 1
Parameterized computational complexity of Dodgson and Young elections
, 2007
"... Abstract. We show that, other than for standard complexity theory with known NPcompleteness results, the computational complexity of the Dodgson and Young election systems is completely different from a parameterized complexity point of view. That is, on the one hand, we present an efficient fixed ..."
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Cited by 21 (7 self)
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Abstract. We show that, other than for standard complexity theory with known NPcompleteness results, the computational complexity of the Dodgson and Young election systems is completely different from a parameterized complexity point of view. That is, on the one hand, we present an efficient fixedparameter algorithm for determining a Condorcet winner in Dodgson elections by a minimum number of switches in the votes. On the other hand, we prove that the corresponding problem for Young elections, where one has to delete votes instead of performing switches, is W[2]complete. In addition, we study Dodgson elections that allow ties between the candidates and give fixedparameter tractability as well as W[2]hardness results depending on the cost model for switching ties. 1
Bidimensionality and Kernels
, 2010
"... Bidimensionality theory appears to be a powerful framework in the development of metaalgorithmic techniques. It was introduced by Demaine et al. [J. ACM 2005] as a tool to obtain subexponential time parameterized algorithms for bidimensional problems on Hminor free graphs. Demaine and Hajiaghayi ..."
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Cited by 21 (12 self)
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Bidimensionality theory appears to be a powerful framework in the development of metaalgorithmic techniques. It was introduced by Demaine et al. [J. ACM 2005] as a tool to obtain subexponential time parameterized algorithms for bidimensional problems on Hminor free graphs. Demaine and Hajiaghayi [SODA 2005] extended the theory to obtain polynomial time approximation schemes (PTASs) for bidimensional problems. In this paper, we establish a third metaalgorithmic direction for bidimensionality theory by relating it to the existence of linear kernels for parameterized problems. In parameterized complexity, each problem instance comes with a parameter k and the parameterized problem is said to admit a linear kernel if there is a polynomial time algorithm, called
Techniques For Practical FixedParameter Algorithms
, 2007
"... The fixedparameter approach is an algorithm design technique for solving combinatorially hard (mostly NPhard) problems. For some of these problems, it can lead to algorithms that are both efficient and yet at the same time guaranteed to find optimal solutions. Focusing on their application to solv ..."
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Cited by 20 (8 self)
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The fixedparameter approach is an algorithm design technique for solving combinatorially hard (mostly NPhard) problems. For some of these problems, it can lead to algorithms that are both efficient and yet at the same time guaranteed to find optimal solutions. Focusing on their application to solving NPhard problems in practice, we survey three main techniques to develop fixedparameter algorithms, namely: kernelization (data reduction with provable performance guarantee), depthbounded search trees and a new technique called iterative compression. Our discussion is circumstantiated by several concrete case studies and provides pointers to various current challenges in the field.
A Duality between Clause Width and Clause Density for SAT
 In IEEE Conference on Computational Complexity (CCC
"... We consider the relationship between the complexities of and those of restricted to formulas of constant density. Let be the infimum of those such that on variables can be decided in time and be the infimum of those such that on variables and clauses can be decided in time. We show that. So, for a ..."
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Cited by 20 (4 self)
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We consider the relationship between the complexities of and those of restricted to formulas of constant density. Let be the infimum of those such that on variables can be decided in time and be the infimum of those such that on variables and clauses can be decided in time. We show that. So, for any, can be solved in time independent of if and only if the same is true for with any fixed density of clauses to variables. We derive some interesting consequences from this. For example, assuming thatis exponentially hard (that is,), of any fixed density can be solved in time whose exponent is strictly less than that for general. We also give an improvement to the sparsification lemma of [12] showing that instances of of density slightly more than exponential in are almost the hardest instances of. The previous result showed this for densities doubly exponential in. 1.
Parameterizing above or below guaranteed values
 J. Comput. System Sci
"... We consider new parameterizations of NPoptimization problems that have nontrivial lower and/or upper bounds on their optimum solution size. The natural parameter, we argue, is the quantity above the lower bound or below the upper bound. We show that for every problem in MAX SNP, the optimum value i ..."
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Cited by 19 (2 self)
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We consider new parameterizations of NPoptimization problems that have nontrivial lower and/or upper bounds on their optimum solution size. The natural parameter, we argue, is the quantity above the lower bound or below the upper bound. We show that for every problem in MAX SNP, the optimum value is bounded below by an unbounded function of the inputsize, and that the aboveguarantee parameterization with respect to this lower bound is fixedparameter tractable. We also observe that approximation algorithms give nontrivial lower or upper bounds on the solution size and that the above or below guarantee question with respect to these bounds is fixedparameter tractable for a subclass of NPoptimization problems. We then introduce the notion of ‘tight ’ lower and upper bounds and exhibit a number of problems for which the aboveguarantee and belowguarantee parameterizations with respect to a tight bound is fixedparameter tractable or Whard. We show that if we parameterize “sufficiently ” above or below the tight bounds, then these parameterized versions are not fixedparameter tractable unless P = NP, for a subclass of NPoptimization problems. We also list several directions to explore in this paradigm. 1
On notions of regularity for data languages
 In FCT
, 2007
"... Motivated by considerations in XML database theory and model checking, data strings have been introduced as an extension of finite alphabet strings which carry, at each position, a symbol and a data value from an infinite domain. Previous work has shown that it is difficult to come up with an expres ..."
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Cited by 19 (4 self)
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Motivated by considerations in XML database theory and model checking, data strings have been introduced as an extension of finite alphabet strings which carry, at each position, a symbol and a data value from an infinite domain. Previous work has shown that it is difficult to come up with an expressive yet decidable automaton model for data languages. Recently, such a model, data automata, was introduced. This paper introduces a simpler but equivalent model and investigates its expressive power, algorithmic and closure properties, and some extensions. 1
Width parameters beyond treewidth and their applications
 Computer Journal
, 2007
"... Besides the very successful concept of treewidth (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth. These are special issues on Parameterized Complexity]), many concepts and parameters measuring the similarity or dissimilarity of structures compare ..."
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Cited by 18 (0 self)
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Besides the very successful concept of treewidth (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth. These are special issues on Parameterized Complexity]), many concepts and parameters measuring the similarity or dissimilarity of structures compared to trees have been born and studied over the past years. These concepts and parameters have proved to be useful tools in many applications, especially in the design of efficient algorithms. Our presented novel look at the contemporary developments of these ‘width ’ parameters in combinatorial structures delivers—besides traditional treewidth and derived dynamic programming schemes—also a number of other useful parameters like branchwidth, rankwidth (cliquewidth) or hypertreewidth. In this contribution, we demonstrate how ‘width ’ parameters of graphs and generalized structures (such as matroids or hypergraphs), can be used to improve the design of parameterized algorithms and the structural analysis in other applications on an abstract level.