Filters








47,684 Hits in 6.9 sec

Kernelization Lower Bounds by Cross-Composition

Hans L. Bodlaender, Bart M. P. Jansen, Stefan Kratsch
2014 SIAM Journal on Discrete Mathematics  
We introduce the cross-composition framework for proving kernelization lower bounds.  ...  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.  ...  We exhibit the power of cross-composition by giving kernelization lower bounds for structural parameterizations of several important graph problems.  ... 
doi:10.1137/120880240 fatcat:xd6ycd7hprax5c2jyhmx47awba

Kernelization Lower Bounds By Cross-Composition [article]

Hans L. Bodlaender and Bart M. P. Jansen and Stefan Kratsch
2012 arXiv   pre-print
We introduce the cross-composition framework for proving kernelization lower bounds.  ...  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.  ...  We exhibit the power of cross-composition by giving kernelization lower bounds for structural parameterizations of several important graph problems.  ... 
arXiv:1206.5941v1 fatcat:drlq6h3bfve3xa5pivdzm4hfga

Cross-Composition: A New Technique for Kernelization Lower Bounds

Hans L. Bodlaender, Bart M. P. Jansen, Stefan Kratsch, Marc Herbstritt
2011 Symposium on Theoretical Aspects of Computer Science  
We introduce a new technique for proving kernelization lower bounds, called cross-composition.  ...  We show its applicability by proving kernelization lower bounds for a number of important graphs problems with structural (non-standard) parameterizations, e.g., Chromatic Number, Clique, and Weighted  ...  How Cross-compositions Yield Lower Bounds The purpose of this section is to prove that cross-compositions yield kernelization lower bounds.  ... 
doi:10.4230/lipics.stacs.2011.165 dblp:conf/stacs/BodlaenderJK11 fatcat:llt6dsbrjbddnoff2ugw5nmqay

Cross-Composition: A New Technique for Kernelization Lower Bounds [article]

Hans L. Bodlaender and Bart M. P. Jansen and Stefan Kratsch
2010 arXiv   pre-print
We introduce a new technique for proving kernelization lower bounds, called cross-composition.  ...  We show its applicability by proving kernelization lower bounds for a number of important graphs problems with structural (non-standard) parameterizations, e.g., Chromatic Number, Clique, and Weighted  ...  How Cross-compositions Imply Lower Bounds The purpose of this section is to prove that cross-compositions imply kernelization lower bounds.  ... 
arXiv:1011.4224v2 fatcat:yqvslhs6cndz3fa5zusrfzsqzu

Lossy kernelization

Daniel Lokshtanov, Fahad Panolan, M. S. Ramanujan, Saket Saurabh
2017 Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2017  
unless NP ⊆ coNP/Poly Our approximate kernels simultaneously beat both the lower bounds on the (normal) kernel size, and the hardness of approximation lower bounds for all three problems.  ...  On the negative side we prove that L P parameterized by the length of the path and S C parameterized by the universe size do not admit even an α-approximate kernel of polynomial size, for any α ≥ 1, unless  ...  In order to prove lower bounds on the size of α-approximate kernels, we generalize the notion of cross compositions to α-gap cross compositions, which are hybrid of cross compositions and gap creating  ... 
doi:10.1145/3055399.3055456 dblp:conf/stoc/LokshtanovPRS17 fatcat:tfxcsrgswjdvjkiqo6mmdhnypm

FPT is Characterized by Useful Obstruction Sets [article]

Michael R. Fellows, Bart M. P. Jansen
2013 arXiv   pre-print
Our new characterization of FPT has a strong connection to the theory of kernelization, as we prove that problems with polynomial kernels can be characterized by obstruction sets whose elements have polynomial  ...  We show how exponential-size minor-minimal obstructions for pathwidth k form the crucial ingredient in a novel OR-cross-composition for k-Pathwidth, complementing the trivial AND-composition that is known  ...  We use the framework of cross-composition to prove kernel lower bounds, including the definition of a polynomial equivalence relation and a cross-composition as given by Bodlaender et al. [3] .  ... 
arXiv:1305.3102v1 fatcat:n2kq4ni6pnf7lnyvuzhkbatydy

FPT Is Characterized by Useful Obstruction Sets [chapter]

Michael R. Fellows, Bart M. P. Jansen
2013 Lecture Notes in Computer Science  
We show how exponential-size minor-minimal obstructions for pathwidth k form the crucial ingredient in a novel or-cross-composition for k-Pathwidth, complementing the trivial and-composition that is known  ...  Our new characterization of FPT has a strong connection to the theory of kernelization, as we prove that problems with polynomial kernels can be characterized by obstruction sets whose elements have polynomial  ...  We use the framework of cross-composition to prove kernel lower bounds, including the definition of a polynomial equivalence relation and a cross-composition as given by Bodlaender et al. [3] .  ... 
doi:10.1007/978-3-642-45043-3_23 fatcat:przvijj6hbbyha3cur3yswagrm

Kernelization – Preprocessing with a Guarantee [chapter]

Daniel Lokshtanov, Neeldhara Misra, Saket Saurabh
2012 Lecture Notes in Computer Science  
We also outline the framework of Bodlaender et al. [9] and Fortnow and Santhanam [38] for showing kernelization lower bounds under reasonable assumptions from classical complexity theory, and highlight  ...  We give an overview of some of the early work in the area and also survey newer techniques that have emerged in the design and analysis of kernelization algorithms.  ...  Having established what a cross-composition algorithm is, and why it implies kernel lower bounds, we now state some applications of this technique.  ... 
doi:10.1007/978-3-642-30891-8_10 fatcat:ihw5qxdnffdybk4f2c4lhds3pe

On Polynomial Kernels for Sparse Integer Linear Programs [article]

Stefan Kratsch
2013 arXiv   pre-print
However, this lower bound only applies for the case when constraints may include an arbitrary number of variables since it follows from lower bounds for Satisfiability and Hitting Set, whose bounded arity  ...  Thus, by a folklore argument, any such ILP admits a kernelization to an equivalent instance of size O(c^n^3).  ...  Otherwise, if the range of each variable is polynomially bounded in n then we establish a polynomial kernelization.  ... 
arXiv:1302.3494v2 fatcat:v3ev2eqlsjenfd7xtkju2wua6q

Recent developments in kernelization: A survey

Stefan Kratsch
2014 Bulletin of the European Association for Theoretical Computer Science  
There is a nice hardness theory, based on the complexity theoretic assumption coNP NP/poly, which can be used to prove lower bounds for kernel size.  ...  Kernelization is a flourishing area of parameterized complexity with many recent results (both upper and lower bounds). Stefan Kratsch shares with us some of the latest developments in the field.  ...  Definition 4 (and/or-cross-composition of bounded cost [12] ).  ... 
dblp:journals/eatcs/Kratsch14 fatcat:7hpcf4kyvvgi5ku3iueodwrpku

Optimal Sparsification for Some Binary CSPs Using Low-Degree Polynomials

Bart M. P. Jansen, Astrid Pieterse
2019 ACM Transactions on Computation Theory  
Upper and lower bounds are established using the concept of kernelization.  ...  Our lower bounds show that not just the number of constraints, but also the encoding size of individual constraints plays an important role.  ...  By Theorem 1, a degree-d cross-composition can be used to rule out generalized kernels of size O(k d−ε ).  ... 
doi:10.1145/3349618 fatcat:p7f56ucksrbczgjnlz63fdf67m

Weak Compositions and Their Applications to Polynomial Lower Bounds for Kernelization [chapter]

Danny Hermelin, Xi Wu
2012 Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms  
We show two examples of weak composition and derive polynomial kernelization lower bounds for d-Bipartite Regular Perfect Code and d-Dimensional Matching, parameterized by the solution size k.  ...  We introduce a new form of composition called weak composition that allows us to obtain polynomial kernelization lower-bounds for several natural parameterized problems.  ...  In particular, Karl provided several insights regarding the main composition algorithm presented in Section 3. We would also like to thank Chandan Saha for referring us to [24] .  ... 
doi:10.1137/1.9781611973099.9 dblp:conf/soda/HermelinW12 fatcat:z4iotnmrhbdsxf7n6yfojcmmqe

Optimal Data Reduction for Graph Coloring Using Low-Degree Polynomials

Bart M. P. Jansen, Astrid Pieterse, Marc Herbstritt
2018 International Symposium on Parameterized and Exact Computation  
Previously, such a lower bound was only known for coloring with q ≥ 4 colors.  ...  First, we use a recent technique of finding redundant constraints by representing them as lowdegree polynomials, to obtain a kernel of bitsize O(k q−1 log k) for q-Coloring parameterized by Vertex Cover  ...  We use the framework of cross-composition [1] to establish kernelization lower bounds, requiring the definitions of polynomial equivalence relations and or-cross-compositions.  ... 
doi:10.4230/lipics.ipec.2017.22 dblp:conf/iwpec/JansenP17 fatcat:w343j3mhxje7xgliz33jfuvkma

Optimal Data Reduction for Graph Coloring Using Low-Degree Polynomials

Bart M. P. Jansen, Astrid Pieterse
2019 Algorithmica  
Previously, such a lower bound was only known for coloring with q ≥ 4 colors.  ...  First, we use a recent technique of finding redundant constraints by representing them as lowdegree polynomials, to obtain a kernel of bitsize O(k q−1 log k) for q-Coloring parameterized by Vertex Cover  ...  We use the framework of cross-composition [1] to establish kernelization lower bounds, requiring the definitions of polynomial equivalence relations and or-cross-compositions.  ... 
doi:10.1007/s00453-019-00578-5 fatcat:wf4yjsxqeje3jgusrf3sujt5ri

On polynomial kernels for sparse integer linear programs

Stefan Kratsch
2016 Journal of computer and system sciences (Print)  
However, this lower bound only applies for the case when constraints may include an arbitrary number of variables since it follows from lower bounds for SAT and Hitting Set, whose bounded arity variants  ...  Thus, by a folklore argument, any such ILP admits a kernelization to an equivalent instance of size O(c n 3 ).  ...  Almost all lower bounds for kernelization apply also for this weaker notion. For our lower bound proof we use the concept of an (or-)cross-composition of Bodlaender et al.  ... 
doi:10.1016/j.jcss.2015.12.002 fatcat:7fqcmj7fevfkbm7o2jxzbvubwa
« Previous Showing results 1 — 15 out of 47,684 results