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31
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 143 (17 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.
Kernelization: New Upper and Lower Bound Techniques
 In Proc. of the 4th International Workshop on Parameterized and Exact Computation (IWPEC), volume 5917 of LNCS
, 2009
"... Abstract. In this survey, we look at kernelization: algorithms that transform in polynomial time an input to a problem to an equivalent input, whose size is bounded by a function of a parameter. Several results of recent research on kernelization are mentioned. This survey looks at some recent resu ..."
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Cited by 54 (0 self)
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Abstract. In this survey, we look at kernelization: algorithms that transform in polynomial time an input to a problem to an equivalent input, whose size is bounded by a function of a parameter. Several results of recent research on kernelization are mentioned. This survey looks at some recent results where a general technique shows the existence of kernelization algorithms for large classes of problems, in particular for planar graphs and generalizations of planar graphs, and recent lower bound techniques that give evidence that certain types of kernelization algorithms do not exist.
Fixedparameter algorithms for artificial intelligence, constraint satisfaction, and database problems
, 2007
"... We survey the parameterized complexity of problems that arise in artificial intelligence, database theory and automated reasoning. In particular, we consider various parameterizations of the constraint satisfaction problem, the evaluation problem of Boolean conjunctive database queries and the propo ..."
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Cited by 32 (10 self)
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We survey the parameterized complexity of problems that arise in artificial intelligence, database theory and automated reasoning. In particular, we consider various parameterizations of the constraint satisfaction problem, the evaluation problem of Boolean conjunctive database queries and the propositional satisfiability problem. Furthermore, we survey parameterized algorithms for problems arising in the context of the stable model semantics of logic programs, for a number of other problems of nonmonotonic reasoning, and for the computation of cores in data exchange.
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 23 (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.
KERNEL(S) FOR PROBLEMS WITH NO KERNEL: ON OUTTREES WITH MANY LEAVES (EXTENDED ABSTRACT)
 STACS 2009
, 2009
"... The kLeaf OutBranching problem is to find an outbranching, that is a rooted oriented spanning tree, with at least k leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms. Here, we take a kernelization based approach to the kLea ..."
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Cited by 19 (7 self)
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The kLeaf OutBranching problem is to find an outbranching, that is a rooted oriented spanning tree, with at least k leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms. Here, we take a kernelization based approach to the kLeafOutBranching problem. We give the first polynomial kernel for Rooted kLeafOutBranching, a variant of kLeafOutBranching where the root of the tree searched for is also a part of the input. Our kernel has cubic size and is obtained using extremal combinatorics. For the kLeafOutBranching problem, we show that no polynomial kernel is possible unless the polynomial hierarchy collapses to third level by applying a recent breakthrough result by Bodlaender et al. (ICALP 2008) in a nontrivial fashion. However, our positive results for Rooted kLeafOutBranching immediately imply that the seemingly intractable kLeafOutBranching problem admits a data reduction to n independent O(k³) kernels. These two results, tractability and intractability side by side, are the first ones separating manytoone kernelization from Turing kernelization. This answers affirmatively an open problem regarding “cheat kernelization” raised by Mike Fellows and Jiong Guo independently.
Algorithmic Metatheorems for Restrictions of Treewidth
"... Possibly the most famous algorithmic metatheorem is Courcelle’s theorem, which states that all MSOexpressible graph properties are decidable in linear time for graphs of bounded treewidth. Unfortunately, the running time’s dependence on the formula describing the problem is in general a tower of e ..."
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Cited by 19 (4 self)
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Possibly the most famous algorithmic metatheorem is Courcelle’s theorem, which states that all MSOexpressible graph properties are decidable in linear time for graphs of bounded treewidth. Unfortunately, the running time’s dependence on the formula describing the problem is in general a tower of exponentials of unbounded height, and there exist lower bounds proving that this cannot be improved even if we restrict ourselves to deciding FO logic on trees. We investigate whether this parameter dependence can be improved by focusing on two proper subclasses of the class of bounded treewidth graphs: graphs of bounded vertex cover and graphs of bounded maxleaf number. We prove stronger algorithmic metatheorems for these more restricted classes of graphs. More specifically, we show it is possible to decide any FO property in both of these classes with a singly exponential parameter dependence and that it is possible to decide MSO logic on graphs of bounded vertex cover with a doubly exponential parameter dependence. We also prove lower bound results which show that our upper bounds cannot be improved significantly, under widely believed complexity assumptions. Our work addresses an open problem posed by Michael Fellows.
Compression via Matroids: A Randomized Polynomial Kernel for Odd Cycle Transversal
"... The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite by deleting at most k of its vertices. In a breakthrough result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a O(4 k kmn) time algorithm for it, the first algorithm with polynomial runtime of ..."
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Cited by 19 (4 self)
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The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite by deleting at most k of its vertices. In a breakthrough result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a O(4 k kmn) time algorithm for it, the first algorithm with polynomial runtime of uniform degree for every fixed k. It is known that this implies a polynomialtime compression algorithm that turns OCT instances into equivalent instances of size at most O(4 k), a socalled kernelization. Since then the existence of a polynomial kernel for OCT, i.e., a kernelization with size bounded polynomially in k, has turned into one of the main open questions in the study of kernelization. Despite the impressive progress in the area, including the recent development of lower bound techniques (Bodlaender
FixedParameter Tractability Results for FullDegree Spanning Tree and Its Dual
 In Proc. the 2nd International Workshop on Parameterized and Exact Computation (IWPEC), Springer LNCS
, 2006
"... We provide firsttime fixedparameter tractability results for the NPhard problems Maximum FullDegree Spanning Tree and MinimumVertex Feedback Edge Set. These problems are dual to each other: In Maximum FullDegree Spanning Tree, the task is to find a spanning tree for a given graph that maximizes ..."
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Cited by 13 (2 self)
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We provide firsttime fixedparameter tractability results for the NPhard problems Maximum FullDegree Spanning Tree and MinimumVertex Feedback Edge Set. These problems are dual to each other: In Maximum FullDegree Spanning Tree, the task is to find a spanning tree for a given graph that maximizes the number of vertices that preserve their degree. For MinimumVertex Feedback Edge Set, the task is to minimize the number of vertices that end up with a reduced degree. Parameterized by the solution size, we exhibit that MinimumVertex Feedback Edge Set is fixedparameter tractable and has a problem kernel with the number of vertices linearly depending on the parameter k. Our main contribution for Maximum FullDegree Spanning Tree, which is W[1]hard, is a linearsize problem kernel when restricted to planar graphs. Moreover, we present a dynamic programming algorithm for graphs of bounded treewidth. Keywords: Fixedparameter tractability, Problem kernel, Data reduction,
A new algorithm for finding trees with many leaves
, 2001
"... We present an algorithm that finds trees with at least k leaves in undirected and directed graphs. These problems are known as Maximum Leaf Spanning Tree for undirected graphs, and, respectively, Directed Maximum Leaf OutTree and Directed Maximum Leaf Spanning OutTree in the case of directed grap ..."
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Cited by 13 (2 self)
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We present an algorithm that finds trees with at least k leaves in undirected and directed graphs. These problems are known as Maximum Leaf Spanning Tree for undirected graphs, and, respectively, Directed Maximum Leaf OutTree and Directed Maximum Leaf Spanning OutTree in the case of directed graphs. The run time of our algorithm is O(poly(V ) + 4 k k 2) on undirected graphs, and O(4 k V ·E) on directed graphs. Currently, the fastest algorithms for these problems have run times of O(poly(n) + 6.75 k poly(k)) and 2 O(k log k) poly(n), respectively.
Spanning trees with many leaves in graphs without diamonds and blossoms
 IN: PROC. 8TH LATIN, BÚZIOS
, 2008
"... It is known that graphs on n vertices with minimum degree at least 3 have spanning trees with at least n/4 + 2 leaves and that this can be improved to (n + 4)/3 for cubic graphs without the diamond K4 − e as a subgraph. We generalize the second result by proving that every graph with minimum degree ..."
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Cited by 11 (3 self)
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It is known that graphs on n vertices with minimum degree at least 3 have spanning trees with at least n/4 + 2 leaves and that this can be improved to (n + 4)/3 for cubic graphs without the diamond K4 − e as a subgraph. We generalize the second result by proving that every graph with minimum degree at least 3, without diamonds and certain subgraphs called blossoms, has a spanning tree with at least (n+4)/3 leaves, and generalize this further by allowing vertices of lower degree. We show that it is necessary to exclude blossoms in order to obtain a bound of the form n/3 + c. We use the new bound to obtain a simple FPT algorithm, which decides in O(m) + O ∗ (6.75 k) time whether a graph of size m has a spanning tree with at least k leaves. This improves the best known time complexity for Max Leaf Spanning Tree.