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329
Incompressibility through Colors and IDs
"... In parameterized complexity each problem instance comes with a parameter k and the parameterized problem is said to admit a polynomial kernel if there are polynomial time preprocessing rules that reduce the input instance down to an instance with size polynomial in k. Many problems have been shown t ..."
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Cited by 24 (5 self)
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In parameterized complexity each problem instance comes with a parameter k and the parameterized problem is said to admit a polynomial kernel if there are polynomial time preprocessing rules that reduce the input instance down to an instance with size polynomial in k. Many problems have been shown to admit polynomial kernels, but it is only recently that a framework for showing the nonexistence of polynomial kernels for specific problems has been developed by Bodlaender et al. [6] and Fortnow and Santhanam [15]. With few exceptions, all known kernelization lower bounds result have been obtained by directly applying this framework. In this paper we show how to combine these results with combinatorial reductions which use colors and IDs in order to prove kernelization lower bounds for a variety of basic problems. Below we give a summary of our main results. All our results are under the assumption that the polynomial hierarchy does not collapse to the third level. • We show that the Steiner Tree problem parameterized by the number of terminals and solution size, and the Connected Vertex Cover and Capacitated Vertex Cover problems do not admit a polynomial kernel. The two latter results are surprising because the closely related Vertex Cover problem admits a kernel of size 2k.
Reflections on multivariate algorithmics and problem parameterization
 In Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS ’10), volume 5 of LIPIcs
"... Abstract. 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 investiga ..."
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Cited by 24 (19 self)
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Abstract. 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.
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 23 (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
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 22 (9 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 22 (3 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.
Combinatorial Optimization on Graphs of Bounded Treewidth
, 2007
"... There are many graph problems that can be solved in linear or polynomial time with a dynamic programming algorithm when the input graph has bounded treewidth. For combinatorial optimization problems, this is a useful approach for obtaining fixedparameter tractable algorithms. Starting from trees an ..."
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Cited by 21 (1 self)
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There are many graph problems that can be solved in linear or polynomial time with a dynamic programming algorithm when the input graph has bounded treewidth. For combinatorial optimization problems, this is a useful approach for obtaining fixedparameter tractable algorithms. Starting from trees and seriesparallel graphs, we introduce the concepts of treewidth and tree decompositions, and illustrate the technique with the Weighted Independent Set problem as an example. The paper surveys some of the latest developments, putting an emphasis on applicability, on algorithms that exploit tree decompositions, and on algorithms that determine or approximate treewidth and find tree decompositions with optimal or close to optimal treewidth. Directions for further research and suggestions for further reading are also given.
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 (11 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
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
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 20 (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
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 19 (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.