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Kernelization of Packing Problems [chapter]

Holger Dell, Dániel Marx
2012 Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms  
Under the same hypothesis, we show tight lower bounds for the kernelization of d-Set Matching and other packing problems.  ...  Obtaining tight bounds for graph packing problems is challenging: We make first progress in this direction by showing non-trivial kernels with O(k^2.5) edges for the problem of finding k vertex-disjoint  ...  We would like to thank Martin Grohe and Dieter van Melkebeek for valuable comments on previous versions of this paper.  ... 
doi:10.1137/1.9781611973099.6 dblp:conf/soda/DellM12 fatcat:nbclqme3dzdvffolelibcbrtke

Linear-Vertex Kernel for the Problem of Packing r-Stars into a Graph without Long Induced Paths [article]

Florian Barbero, Gregory Gutin, Mark Jones, Bin Sheng, Anders Yeo
2015 arXiv   pre-print
We study the problem of packing $k$ vertex-disjoint copies of $K_{1,r}$ ($k\ge 2$) into a graph $G$ from parameterized preprocessing, i.e., kernelization, point of view.  ...  Let ${\cal G}_d$ be the set of graphs with no induced path on $d$ vertices.  ...  Proof of Theorem 1 Note that the 1-star packing problem is the classic maximum matching problem and if k = 1, the r-star packing problem is equivalent to deciding whether ∆(G) ≥ r.  ... 
arXiv:1510.03564v1 fatcat:5mh35bb5wvbpdeefubm5wxx3gi

Subquadratic Kernels for Implicit 3-Hitting Set and 3-Set Packing Problems [chapter]

Tien-Nam Le, Daniel Lokshtanov, Saket Saurabh, Stéphan Thomassé, Meirav Zehavi
2018 Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms  
In fact, such kernels can be obtained by interpreting these problems as finding either a packing of k pairwise disjoint sets of size 3 (3-Set Packing) or a hitting set of size at most k for a family of  ...  Our results resolve an open problem from WorKer 2010 on the existence of kernels with O(k 2− ) vertices for FVST and CVD.  ...  Finally, we say a few words about our kernels for packing problems, that is, for TPT and Induced P 3 -Packing.  ... 
doi:10.1137/1.9781611975031.23 dblp:conf/soda/LeL0TZ18 fatcat:so2ctg3hcnc35dfi72xkoxf27i

A Problem Kernelization for Graph Packing [chapter]

Hannes Moser
2009 Lecture Notes in Computer Science  
Our approach combines ideas from an improved kernelization of Hitting Set [1] and from problem kernels for generalized matching and set packing problems [10] to achieve this.  ...  However, to the best of our knowledge, there is no example of a problem that does admit a polynomial-size kernel, but no linear-size kernel.  ...  Triangle Packing has a problem kernel with at most 45k 2 vertices. Proof.  ... 
doi:10.1007/978-3-540-95891-8_37 fatcat:kcw6ixvbyzgwjnfhc64hmjbqw4

Grassmannian Packings in Neural Networks: Learning with Maximal Subspace Packings for Diversity and Anti-Sparsity [article]

Dian Ang Yap, Nicholas Roberts, Vinay Uday Prabhu
2019 arXiv   pre-print
Drawing inspiration from information theory and wireless communications, we demonstrate the intersection of coding theory and deep learning through the Grassmannian subspace packing problem in CNNs.  ...  Kernel sparsity ("dying ReLUs") and lack of diversity are commonly observed in CNN kernels, which decreases model capacity.  ...  Acknowledgments We would like to acknowledge Matthew McAteer for helpful discussions and contrbution of designs and plots for Figures 5 and 6 .  ... 
arXiv:1911.07418v1 fatcat:xufyag6rnfctjhbtgeqdphpdie

Kernelization Algorithms for Packing Problems Allowing Overlaps (Extended Version) [article]

Henning Fernau, Alejandro López-Ortiz, Jazmín Romero
2015 arXiv   pre-print
We reduce the $r$-Set Packing with $t$-Membership to a problem kernel with $O((r+1)^r k^{r})$ elements while we achieve a kernel with $O(r^r k^{r-t-1})$ elements for the $r$-Set Packing with $t$-Overlap  ...  We consider the problem of discovering overlapping communities in networks which we model as generalizations of Graph Packing problems with overlap.  ...  We obtain a kernel result for the r-Set Packing with t-Membership problem by running a kernelization algorithm on the transformed instance of the Set Packing problem.  ... 
arXiv:1411.6915v3 fatcat:tsireoh7rnho7chguubnfkhnua

Kernelization of Cycle Packing with Relaxed Disjointness Constraints

Akanksha Agrawal, Daniel Lokshtanov, Diptapriyo Majumdar, Amer E. Mouawad, Saket Saurabh
2018 SIAM Journal on Discrete Mathematics  
While the Pairwise Disjoint Cycle Packing problem admits a polynomial kernel for all t ≥ 1, the kernelization complexity of Almost Disjoint Cycle Packing reveals an interesting spectrum of upper and lower  ...  A key result in the field of kernelization, a subfield of parameterized complexity, states that the classic Disjoint Cycle Packing problem, i.e. finding k vertex disjoint cycles in a given graph G, admits  ...  For any t ≥ 2, we can compress an instance of Pairwise Disjoint Cycle Packing to an equivalent instance of Succinct Pairwise Disjoint Cycle Packing using at most O(k 5 log 2 k) bits.  ... 
doi:10.1137/17m1136614 fatcat:2bb622lgo5fy3i7ooe3at4ybse

Looking at the stars

Elena Prieto, Christian Sloper
2006 Theoretical Computer Science  
In the framework of parameterized complexity, we analyze a particular family of instances of this problem, namely the packing of stars. We give a quadratic kernel for packing k copies of H = K 1,s .  ...  The problem of packing k vertex-disjoint copies of a graph H into another graph G is NP-complete if H has more than two vertices in some connected component.  ...  Acknowledgment We would like to thank Mike Fellows for all the inspiring conversations leading to the completion of this paper.  ... 
doi:10.1016/j.tcs.2005.10.009 fatcat:4weu3kq6ovhq7l3qobylcw5gly

Polynomial Kernels for Weighted Problems [article]

Michael Etscheid, Stefan Kratsch, Matthias Mnich, Heiko Röglin
2015 arXiv   pre-print
Kernelization is a formalization of efficient preprocessing for NP-hard problems using the framework of parameterized complexity.  ...  We further illustrate its applicability by giving polynomial kernels also for weighted versions of several well-studied parameterized problems.  ...  Since these problems generalize quite a few interesting hitting/covering and packing problems, this extends the list of problems whose weighted versions directly benefit from our results.  ... 
arXiv:1507.03439v1 fatcat:35iefk6kyzgnhaeqh3b2z3z5ni

Packing Edge Disjoint Triangles: A Parameterized View [chapter]

Luke Mathieson, Elena Prieto, Peter Shaw
2004 Lecture Notes in Computer Science  
We achieve a kernel size bounded by 4k, where k is the number of triangles in the packing.  ...  The problem of packing k edge-disjoint triangles in a graph has been thoroughly studied both in the classical complexity and the approximation fields and it has a wide range of applications in many areas  ...  This method of coordinatized kernels has already been used to tackle many other graph maximization problems such as k-Nonblocker [DFFPR03] , k-Max Leaf [FMRS00] and k-Star Packing [FPS03] proving  ... 
doi:10.1007/978-3-540-28639-4_12 fatcat:vp5fnhk725eg7lfe6q2j35srtq

Finding k Disjoint Triangles in an Arbitrary Graph [chapter]

Mike Fellows, Pinar Heggernes, Frances Rosamond, Christian Sloper, Jan Arne Telle
2004 Lecture Notes in Computer Science  
We introduce a preprocessing (kernelization) technique based on crown decompositions of an auxiliary graph.  ...  We consider exact algorithms for the NP-complete problem of deciding whether an input graph on n vertices has k vertex-disjoint copies of a fixed graph H.  ...  are: (1) We have shown an O(k 3 ) problem kernel for the problem of packing k K 3 s.  ... 
doi:10.1007/978-3-540-30559-0_20 fatcat:r327erg35vgprhpifoqjbwl5y4

Parameterized algorithms of fundamental NP-hard problems: a survey

Wenjun Li, Yang Ding, Yongjie Yang, R. Simon Sherratt, Jong Hyuk Park, Jin Wang
2020 Human-Centric Computing and Information Sciences  
, Feedback Vertex Set, Hyperplane Cover, Vertex Cover, Packing and Matching problems.  ...  All of these problems have been widely applied in various areas, such as Internet of Things, Wireless Sensor Networks, Artificial Intelligence, Bioinformatics, Big Data, and so on.  ...  Acknowledgements The authors would like to thank the reviewers for their valuable comments which improve the quality of this paper dramatically.  ... 
doi:10.1186/s13673-020-00226-w fatcat:2ncywbdlnrdtzliiz4tyi76df4

Characterizing the easy-to-find subgraphs from the viewpoint of polynomial-time algorithms, kernels, and Turing kernels [chapter]

Bart M. P. Jansen, Dániel Marx
2014 Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms  
We study two fundamental problems related to finding subgraphs: (1) given graphs G and H, Subgraph Test asks if H is isomorphic to a subgraph of G, (2) given graphs G, H, and an integer t, Packing asks  ...  if G contains t vertexdisjoint subgraphs isomorphic to H. do not give a complete characterization of the cases when F-Subgraph Test admits polynomial many-one kernels, but show examples that this question  ...  We also define the more general F-Packing problem, where H is restricted to be in F. Kernels and Turing kernels.  ... 
doi:10.1137/1.9781611973730.42 dblp:conf/soda/JansenM15 fatcat:k67ttnsabjcitiaox6khfgcfpe

Explicit linear kernels for packing problems [article]

Valentin Garnero, Christophe Paul, Ignasi Sau, Dimitrios M. Thilikos
2016 arXiv   pre-print
Finally, we also provide linear kernels for the versions of these problems where one wants to pack subgraphs instead of minors.  ...  [ICALP 2013]) guaranteeing the existence of linear kernels on sparse graphs for problems satisfying some generic conditions.  ...  We also provide linear kernels for the versions of the above problems where one wants to pack subgraphs instead of minors (as one could expect, the kernels for subgraphs are considerably simpler than those  ... 
arXiv:1610.06131v1 fatcat:m2pxlgstgvf3vm6teqxfr3ehsu

Faster Fixed-Parameter Tractable Algorithms for Matching and Packing Problems

M. R. Fellows, C. Knauer, N. Nishimura, P. Ragde, F. Rosamond, U. Stege, D. M. Thilikos, S. Whitesides
2007 Algorithmica  
We first establish a general framework for finding and exploiting small problem kernels (of size polynomial in k). Previously such a kernel was known only for triangle packing.  ...  We obtain faster algorithms for problems such as r-dimensional matching, r-set packing, graph packing, and graph edge packing when the size k of the solution is considered a parameter.  ...  Introduction In this paper we demonstrate a general method for solving parameterized packing and matching problems by first finding a problem kernel, and then showing how color-coding (the use of nice  ... 
doi:10.1007/s00453-007-9146-y fatcat:kxt6holtirftfccq46oluw2yc4
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