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Kernelization of Packing Problems
[chapter]

2012
*
Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms
*

Under the same hypothesis, we show tight lower bounds for the

doi:10.1137/1.9781611973099.6
dblp:conf/soda/DellM12
fatcat:nbclqme3dzdvffolelibcbrtke
*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. ...##
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Linear-Vertex Kernel for the Problem of Packing r-Stars into a Graph without Long Induced Paths
[article]

2015
*
arXiv
*
pre-print

We study the

arXiv:1510.03564v1
fatcat:5mh35bb5wvbpdeefubm5wxx3gi
*problem**of**packing*k vertex-disjoint copies*of*K_1,r (k> 2) into a graph G from parameterized preprocessing, i.e.,*kernelization*, point*of*view. ... Let 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. ...##
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Subquadratic Kernels for Implicit 3-Hitting Set and 3-Set Packing Problems
[chapter]

2018
*
Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms
*

In fact, such

doi:10.1137/1.9781611975031.23
dblp:conf/soda/LeL0TZ18
fatcat:so2ctg3hcnc35dfi72xkoxf27i
*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*. ...##
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Kernelization Algorithms for Packing Problems Allowing Overlaps (Extended Version)
[article]

2015
*
arXiv
*
pre-print

We reduce the r-Set

arXiv:1411.6915v3
fatcat:tsireoh7rnho7chguubnfkhnua
*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*. ...##
###
A Problem Kernelization for Graph Packing
[chapter]

2009
*
Lecture Notes in Computer Science
*

Our approach combines ideas from an improved

doi:10.1007/978-3-540-95891-8_37
fatcat:kcw6ixvbyzgwjnfhc64hmjbqw4
*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. ...##
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Kernelization of Cycle Packing with Relaxed Disjointness Constraints

2018
*
SIAM Journal on Discrete Mathematics
*

While the Pairwise Disjoint Cycle

doi:10.1137/17m1136614
fatcat:2bb622lgo5fy3i7ooe3at4ybse
*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 ... The*kernelization*algorithm presented below is inspired by the lossy*kernel*for the Cycle*Packing**problem*given in [22] . ...##
###
Grassmannian Packings in Neural Networks: Learning with Maximal Subspace Packings for Diversity and Anti-Sparsity
[article]

2019
*
arXiv
*
pre-print

Drawing inspiration from information theory and wireless communications, we demonstrate the intersection

arXiv:1911.07418v1
fatcat:xufyag6rnfctjhbtgeqdphpdie
*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 . ...##
###
Looking at the stars

2006
*
Theoretical Computer Science
*

In the framework

doi:10.1016/j.tcs.2005.10.009
fatcat:4weu3kq6ovhq7l3qobylcw5gly
*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. ...##
###
Packing Edge Disjoint Triangles: A Parameterized View
[chapter]

2004
*
Lecture Notes in Computer Science
*

We achieve a

doi:10.1007/978-3-540-28639-4_12
fatcat:vp5fnhk725eg7lfe6q2j35srtq
*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 ...##
###
Polynomial Kernels for Weighted Problems
[article]

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. ...

##
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Finding k Disjoint Triangles in an Arbitrary Graph
[chapter]

2004
*
Lecture Notes in Computer Science
*

We introduce a preprocessing (

doi:10.1007/978-3-540-30559-0_20
fatcat:r327erg35vgprhpifoqjbwl5y4
*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. ...##
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Parameterized algorithms of fundamental NP-hard problems: a survey

2020
*
Human-Centric Computing and Information Sciences
*

, Feedback Vertex Set, Hyperplane Cover, Vertex Cover,

doi:10.1186/s13673-020-00226-w
fatcat:2ncywbdlnrdtzliiz4tyi76df4
*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. ...##
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Characterizing the easy-to-find subgraphs from the viewpoint of polynomial-time algorithms, kernels, and Turing kernels
[chapter]

2014
*
Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms
*

We study two fundamental

doi:10.1137/1.9781611973730.42
dblp:conf/soda/JansenM15
fatcat:k67ttnsabjcitiaox6khfgcfpe
*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*. ...##
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Faster Fixed-Parameter Tractable Algorithms for Matching and Packing Problems

2007
*
Algorithmica
*

We first establish a general framework for finding and exploiting small

doi:10.1007/s00453-007-9146-y
fatcat:kxt6holtirftfccq46oluw2yc4
*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 ...##
###
Faster Fixed-Parameter Tractable Algorithms for Matching and Packing Problems
[chapter]

2004
*
Lecture Notes in Computer Science
*

We first establish a general framework for finding and exploiting small

doi:10.1007/978-3-540-30140-0_29
fatcat:vg5dk5mdvrfxvnycgjfszlfjg4
*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 ...
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