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Journal ArticleDOI

Kernel bounds for disjoint cycles and disjoint paths

TL;DR: This paper builds upon recent results, that show that NP-complete problems that are 'or-compositional' do not have polynomial kernels, unless NP@?coNP/poly, by adding a notion of transformation.
About: This article is published in Theoretical Computer Science.The article was published on 2011-08-01 and is currently open access. It has received 269 citations till now. The article focuses on the topics: Disjoint sets & Kernelization.
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Book
27 Jul 2015
TL;DR: This comprehensive textbook presents a clean and coherent account of most fundamental tools and techniques in Parameterized Algorithms and is a self-contained guide to the area, providing a toolbox of algorithmic techniques.
Abstract: This comprehensive textbook presents a clean and coherent account of most fundamental tools and techniques in Parameterized Algorithms and is a self-contained guide to the area. The book covers many of the recent developments of the field, including application of important separators, branching based on linear programming, Cut & Count to obtain faster algorithms on tree decompositions, algorithms based on representative families of matroids, and use of the Strong Exponential Time Hypothesis. A number of older results are revisited and explained in a modern and didactic way. The book provides a toolbox of algorithmic techniques. Part I is an overview of basic techniques, each chapter discussing a certain algorithmic paradigm. The material covered in this part can be used for an introductory course on fixed-parameter tractability. Part II discusses more advanced and specialized algorithmic ideas, bringing the reader to the cutting edge of current research. Part III presents complexity results and lower bounds, giving negative evidence by way of W[1]-hardness, the Exponential Time Hypothesis, and kernelization lower bounds. All the results and concepts are introduced at a level accessible to graduate students and advanced undergraduate students. Every chapter is accompanied by exercises, many with hints, while the bibliographic notes point to original publications and related work.

1,544 citations


Cites background from "Kernel bounds for disjoint cycles a..."

  • ...4, point 1 originates in [35], points 3, 9, 10, 17 originate in [141], points 7, 8 originate in [121], points 11, 16, 18 originate in [114], points 12, 13 originate in [56], point 14 originates in [276], point 15 originates in [53], and point 19 originates in [117]....

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01 Jan 2009
TL;DR: For any integer d ≥ 3 and positive real e, it was shown in this article that satisfiability for n-variable d-CNF formulas has a protocol of cost O(nd √ log n−1/e) where n is the number of bits of communication from the first player to the second player.
Abstract: Consider the following two-player communication process to decide a language L: The first player holds the entire input x but is polynomially bounded; the second player is computationally unbounded but does not know any part of x; their goal is to decide cooperatively whether x belongs to L at small cost, where the cost measure is the number of bits of communication from the first player to the second player.For any integer d ≥ 3 and positive real e, we show that, if satisfiability for n-variable d-CNF formulas has a protocol of cost O(nd − e), then coNP is in NP/poly, which implies that the polynomial-time hierarchy collapses to its third level. The result even holds when the first player is conondeterministic, and is tight as there exists a trivial protocol for e = 0. Under the hypothesis that coNP is not in NP/poly, our result implies tight lower bounds for parameters of interest in several areas, namely sparsification, kernelization in parameterized complexity, lossy compression, and probabilistically checkable proofs.By reduction, similar results hold for other NP-complete problems. For the vertex cover problem on n-vertex d-uniform hypergraphs, this statement holds for any integer d ≥ 2. The case d = 2 implies that no NP-hard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of O(k2 − e) edges unless coNP is in NP/poly, where k denotes the size of the deletion set. Kernels consisting of O(k2) edges are known for several problems in the class, including vertex cover, feedback vertex set, and bounded-degree deletion.

225 citations

Posted Content
TL;DR: In this paper, the authors introduce the cross-composition framework for proving kernelization lower bounds, which generalizes and strengthens the recent techniques of using composition algorithms and of transferring the lower bounds via polynomial parameter transformations.
Abstract: We introduce the cross-composition framework for proving kernelization lower bounds. A classical problem L AND/OR-cross-composes into a parameterized problem Q if it is possible to efficiently construct an instance of Q with polynomially bounded parameter value that expresses the logical AND or OR of a sequence of instances of L. Building on work by Bodlaender et al. (ICALP 2008) and using a result by Fortnow and Santhanam (STOC 2008) with a refinement by Dell and van Melkebeek (STOC 2010), we show that if an NP-hard problem OR-cross-composes into a parameterized problem Q then Q does not admit a polynomial kernel unless NP \subseteq coNP/poly and the polynomial hierarchy collapses. Similarly, an AND-cross-composition for Q rules out polynomial kernels for Q under Bodlaender et al.'s AND-distillation conjecture. Our technique generalizes and strengthens the recent techniques of using composition algorithms and of transferring the lower bounds via polynomial parameter transformations. We show its applicability by proving kernelization lower bounds for a number of important graphs problems with structural (non-standard) parameterizations, e.g., Clique, Chromatic Number, Weighted Feedback Vertex Set, and Weighted Odd Cycle Transversal do not admit polynomial kernels with respect to the vertex cover number of the input graphs unless the polynomial hierarchy collapses, contrasting the fact that these problems are trivially fixed-parameter tractable for this parameter. After learning of our results, several teams of authors have successfully applied the cross-composition framework to different parameterized problems. For completeness, our presentation of the framework includes several extensions based on this follow-up work. For example, we show how a relaxed version of OR-cross-compositions may be used to give lower bounds on the degree of the polynomial in the kernel size.

196 citations

Book
28 Feb 2019
TL;DR: Kernelization: Theory of Parameterized Preprocessing, by Fomin et al., is unique in that it is a text focusing solely on the titular topic of kernelization, and is able to more effectively showcase and teach the tools used in the field than a more traditional text on fixed parameter complexity.
Abstract: Preprocessing, or data reduction, is a standard technique for simplifying and speeding up computation. Written by a team of experts in the field, this book introduces a rapidly developing area of preprocessing analysis known as kernelization. The authors provide an overview of basic methods and important results, with accessible explanations of the most recent advances in the area, such as meta-kernelization, representative sets, polynomial lower bounds, and lossy kernelization. The text is divided into four parts, which cover the different theoretical aspects of the area: upper bounds, meta-theorems, lower bounds, and beyond kernelization. The methods are demonstrated through extensive examples using a single data set. Written to be self-contained, the book only requires a basic background in algorithmics and will be of use to professionals, researchers and graduate students in theoretical computer science, optimization, combinatorics, and related fields.

181 citations

Proceedings ArticleDOI
20 Oct 2012
TL;DR: This work applies the representative sets tool to the problem of finding irrelevant vertices in graph cut problems, that is, vertices which can be made undeletable without affecting the status of the problem, and gives the first significant progress towards a polynomial kernel for the Multiway Cut problem.
Abstract: The existence of a polynomial kernel for Odd Cycle Transversal was a notorious open problem in parameterized complexity. Recently, this was settled by the present authors (Kratsch and Wahlstr\"om, SODA 2012), with a randomized polynomial kernel for the problem, using matroid theory to encode flow questions over a set of terminals in size polynomial in the number of terminals (rather than the total graph size, which may be superpolynomially larger). In the current work we further establish the usefulness of matroid theory to kernelization by showing applications of a result on representative sets due to Lov\'asz (Combinatorial Surveys 1977) and Marx (TCS 2009). We show how representative sets can be used to give a polynomial kernel for the elusive Almost 2-sat problem (where the task is to remove at most k clauses to make a 2-CNF formula satisfiable), solving a major open problem in kernelization. We further apply the representative sets tool to the problem of finding irrelevant vertices in graph cut problems, that is, vertices which can be made undeletable without affecting the status of the problem. This gives the first significant progress towards a polynomial kernel for the Multiway Cut problem, in particular, we get a polynomial kernel for Multiway Cut instances with a bounded number of terminals. Both these kernelization results have significant spin-off effects, producing the first polynomial kernels for a range of related problems. More generally, the irrelevant vertex results have implications for covering min-cuts in graphs. In particular, given a directed graph and a set of terminals, we can find a set of size polynomial in the number of terminals (a cut-covering set) which contains a minimum vertex cut for every choice of sources and sinks from the terminal set. Similarly, given an undirected graph and a set of terminals, we can find a set of vertices, of size polynomial in the number of terminals, which contains a minimum multiway cut for every partition of the terminals into a bounded number of sets. Both results are polynomial time. We expect this to have further applications, in particular, we get direct, reduction rule-based kernelizations for all problems above, in contrast to the indirect compression-based kernel previously given for Odd Cycle Transversal. All our results are randomized, with failure probabilities which can be made exponentially small in the size of the input, due to needing a representation of a matroid to apply the representative sets tool.

179 citations


Cites methods from "Kernel bounds for disjoint cycles a..."

  • ...[39]), we get a polynomial coRP-kernelization for DIGRAPH PAIR CUT....

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References
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Book
01 Jan 1979
TL;DR: The second edition of a quarterly column as discussed by the authors provides a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness,” W. H. Freeman & Co., San Francisco, 1979.
Abstract: This is the second edition of a quarterly column the purpose of which is to provide a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’’ W. H. Freeman & Co., San Francisco, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed. Readers having results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.), or open problems they would like publicized, should send them to David S. Johnson, Room 2C355, Bell Laboratories, Murray Hill, NJ 07974, including details, or at least sketches, of any new proofs (full papers are preferred). In the case of unpublished results, please state explicitly that you would like the results mentioned in the column. Comments and corrections are also welcome. For more details on the nature of the column and the form of desired submissions, see the December 1981 issue of this journal.

40,020 citations

Book
06 Nov 1998
TL;DR: An approach to complexity theory which offers a means of analysing algorithms in terms of their tractability, and introduces readers to new classes of algorithms which may be analysed more precisely than was the case until now.
Abstract: An approach to complexity theory which offers a means of analysing algorithms in terms of their tractability. The authors consider the problem in terms of parameterized languages and taking "k-slices" of the language, thus introducing readers to new classes of algorithms which may be analysed more precisely than was the case until now. The book is as self-contained as possible and includes a great deal of background material. As a result, computer scientists, mathematicians, and graduate students interested in the design and analysis of algorithms will find much of interest.

3,651 citations


"Kernel bounds for disjoint cycles a..." refers background in this paper

  • ..., [9,10,11]) are similar, but allow non-polynomial growth of the parameter, and are used for a different purpose: to show hardness for W [1] or a related class....

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Journal ArticleDOI
TL;DR: An algorithm, which for fixed k ≥ 0 has running time O (| V(G) | 3 ), to solve the following problem: given a graph G and k pairs of vertices of G, decide if there are k mutually vertex-disjoint paths of G joining the pairs.

1,438 citations