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Constraint satisfaction problems parameterized above or below tight bounds: A survey
 The Multivariate Algorithmic Revolution and Beyond, volume 7370 of Lecture Notes in Computer Science
, 2012
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Kernelization Lower Bounds through Colors and IDs
, 2009
"... In parameterized complexity each problem instance comes with a parameter k, and a parameterized problem is said to admit a polynomial kernel if there are polynomial time preprocessing rules that reduce the input instance to an instance with size polynomial in k. Many problems have been shown to admi ..."
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In parameterized complexity each problem instance comes with a parameter k, and a parameterized problem is said to admit a polynomial kernel if there are polynomial time preprocessing rules that reduce the input instance 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 [17]. 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 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 k, and the CONNECTED VERTEX COVER and CAPACITATED VERTEX COVER problems do not admit a polynomial kernel. The two latter results are surpris
On the Directed DegreePreserving Spanning Tree Problem
"... Abstract. In this paper we initiate a systematic study of the Reduced Degree Spanning Tree problem, where given a digraph D and a nonnegative integer k, the goal is to construct a spanning outtree with at most k vertices of reduced outdegree. This problem is a directed analog of the wellstudied Mi ..."
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Abstract. In this paper we initiate a systematic study of the Reduced Degree Spanning Tree problem, where given a digraph D and a nonnegative integer k, the goal is to construct a spanning outtree with at most k vertices of reduced outdegree. This problem is a directed analog of the wellstudied MinimumVertex Feedback Edge Set problem. We show that this problem is fixedparameter tractable and admits a problem kernel with at most 8k vertices on strongly connected digraphs and O(k 2) vertices on general digraphs. We also give an algorithm for this problem on general digraphs with runtime O ∗ (5.942 k). This adds the Reduced Degree Spanning Tree problem to the small list of directed graph problems for which fixedparameter tractable algorithms are known. Finally, we consider the dual of Reduced Degree Spanning Tree, that is, given a digraph D and a nonnegative integer k, the goal is to construct a spanning outtree of D with at least k vertices of full outdegree. We show that this problem is W[1]hard on two important digraph classes: directed acyclic graphs and strongly connected digraphs. 1
Limits of preprocessing
 In Proceedings of the TwentyFifth Conference on Artificial Intelligence, AAAI 2011
, 2011
"... We present a first theoretical analysis of the power of polynomialtime preprocessing for important combinatorial problems from various areas in AI. We consider problems from Constraint Satisfaction, Global Constraints, Satisfiability, Nonmonotonic and Bayesian Reasoning. We show that, subject to a ..."
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We present a first theoretical analysis of the power of polynomialtime preprocessing for important combinatorial problems from various areas in AI. We consider problems from Constraint Satisfaction, Global Constraints, Satisfiability, Nonmonotonic and Bayesian Reasoning. We show that, subject to a complexity theoretic assumption, none of the considered problems can be reduced by polynomialtime preprocessing to a problem kernel whose size is polynomial in a structural problem parameter of the input, such as induced width or backdoor size. Our results provide a firm theoretical boundary for the performance of polynomialtime preprocessing algorithms for the considered problems.
Note on Maximal Bisection above Tight Lower Bound
"... In a graph G = (V, E), a bisection (X, Y) is a partition of V into sets X and Y such that X  ≤ Y  ≤ X+1. The size of (X, Y) is the number of edges between X and Y. In the Max Bisection problem we are given a graph G = (V, E) and are required to find a bisection of maximum size. It is not har ..."
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In a graph G = (V, E), a bisection (X, Y) is a partition of V into sets X and Y such that X  ≤ Y  ≤ X+1. The size of (X, Y) is the number of edges between X and Y. In the Max Bisection problem we are given a graph G = (V, E) and are required to find a bisection of maximum size. It is not hard to see that ⌈E/2 ⌉ is a tight lower bound on the maximum size of a bisection of G. We study parameterized complexity of the following parameterized problem called Max Bisection above Tight Lower Bound (MaxBisecATLB): decide whether a graph G = (V, E) has a bisection of size at least ⌈E/2 ⌉ + k, where k is the parameter. We show that this parameterized problem has a kernel with O(k²) vertices and O(k³) edges, i.e., every instance of MaxBisecATLB is equivalent to an instance of MaxBisecATLB on a graph with at most O(k²) vertices and O(k³) edges.
On the Directed Full Degree Spanning Tree Problem
"... Abstract. We study the parameterized complexity of a directed analog of the Full Degree Spanning Tree problem where, given a digraph D and a nonnegative integer k, the goal is to construct a spanning outtree T of D such that at least k vertices in T have the same outdegree as in D. We show that t ..."
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Abstract. We study the parameterized complexity of a directed analog of the Full Degree Spanning Tree problem where, given a digraph D and a nonnegative integer k, the goal is to construct a spanning outtree T of D such that at least k vertices in T have the same outdegree as in D. We show that this problem is W[1]hard even on the class of directed acyclic graphs. In the dual version, called Reduced Degree Spanning Tree, one is required to construct a spanning outtree T such that at most k vertices in T have outdegrees that are different from that in D. We show that this problem is fixedparameter tractable and that it admits a problem kernel with at most 8k vertices on strongly connected digraphs and O(k 2 ) vertices on general digraphs. We also give an algorithm for this problem on general digraphs with running time O(5.942 k · n O(1) ), where n is the number of vertices in the input digraph.