A Problem Kernelization for Graph Packing

Moser, Hannes

doi:10.1007/978-3-540-95891-8_37

Cited by 26 publications

(19 citation statements)

References 27 publications

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“…In addition to the quadratic kernel for Triangle Vertex Deletion, our 3HS algorithm has been used to deduce quadratic-order kernels for Cluster Vertex Deletion and Feedback Vertex Set in tournaments, which were studied in [12]. Moreover, the approach introduced in this paper has been used in kernelization algorithms for some maximization problems such as Triangle Packing, 3-Set Packing and 3D Matching [4,15].…”

Section: Discussionmentioning

confidence: 99%

A kernelization algorithm for d-Hitting Set

Abu-Khzam¹

2010

Journal of Computer and System Sciences

139

215

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For a given parameterized problem, π , a kernelization algorithm is a polynomial-time pre-processing procedure that transforms an arbitrary instance of π into an equivalent one whose size depends only on the input parameter(s). The resulting instance is called a problem kernel. In this paper, a kernelization algorithm for the 3-Hitting Set problem is presented along with a general kernelization for d-Hitting Set. For 3-Hitting Set, an arbitrary instance is reduced into an equivalent one that contains at most 5k 2 +k elements.This kernelization is an improvement over previously known methods that guarantee cubic-order kernels. Our method is used also to obtain quadratic kernels for several other problems. For a constant d 3, a kernelization of d-Hitting Set is achieved by a non-trivial generalization of the 3-Hitting Set method, and guarantees a kernel whose order does not exceed (2d − 1)k d−1 + k.

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Section: Discussionmentioning

confidence: 99%

A kernelization algorithm for d-Hitting Set

Abu-Khzam¹

2010

Journal of Computer and System Sciences

139

215

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“…Inspired by the ideas in Fellows et al [2008] and Moser [2009] in Section 5, we give a kernelization algorithm that reduces r-Set Packing with t-Overlap to a kernel with O(r r k r−t−1 ) elements. By transforming an instance of H-Packing with t-Overlap to an instance of r-Set Packing with t-Overlap, we achieve a kernel with O(r r k r−t−1 ) vertices for the H-Packing with t-Overlap problem and for its induced version as well.…”

Section: Introductionmentioning

confidence: 99%

“…Using polynomial parameter transformations, we reduce the r-Set Packing with t-Membership to a problem kernel with O((r + 1) r k r ) elements and the H-Packing with t-Membership and its edge version to problem kernels with O((r + 1) r k r ) and O((m + 1) m k m ) vertices, respectively. On the other hand, by generalizing [Fellows et al 2008;Moser 2009], we achieve a kernel with O(r r k r−t−1 ) elements for the r-Set Packing with t-Overlap and kernels with O(r r k r−t−1 ) and O(m m k m−t−1 ) vertices for the H-Packing with t-Overlap and its edge version, respectively. In all cases, k is the input parameter, while t, r, and m are constants.…”

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confidence: 99%

Using Parametric Transformations Toward Polynomial Kernels for Packing Problems Allowing Overlaps

Fernau

López-Ortíz

Romero

2015

ACM Trans. Comput. Theory

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We consider the problem of discovering overlapping communities in networks that we model as generalizations of the Set and Graph Packing problems with overlap. As usual for Set Packing problems, we seek a collection S ⊆ S consisting of at least k sets subject to certain disjointness restrictions. In the r-Set Packing with t-Membership, each element of U belongs to at most t sets of S , while in r-Set Packing with t-Overlap, each pair of sets in S overlaps in at most t elements. For both problems, each set of S has at most r elements.Similarly, both of our Graph Packing problems seek a collection K of at least k subgraphs in a graph G, each isomorphic to a graph H ∈ H. In H-Packing with t-Membership, each vertex of G belongs to at most t subgraphs of K, while in H-Packing with t-Overlap, each pair of subgraphs in K overlaps in at most t vertices. For both problems, each member of H has at most r vertices and m edges, where t, r, and m are constants.Here, we show NP-completeness results for all of our packing problems. Furthermore, we give a dichotomy result for the H-Packing with t-Membership problem analogous to the Kirkpatrick and Hell dichotomy [Kirkpatrick and Hell 1978]. Using polynomial parameter transformations, we reduce the r-Set Packing with t-Membership to a problem kernel with O((r + 1) r k r ) elements and the H-Packing with t-Membership and its edge version to problem kernels with O((r + 1) r k r ) and O((m + 1) m k m ) vertices, respectively. On the other hand, by generalizing [Fellows et al. 2008;Moser 2009], we achieve a kernel with O(r r k r−t−1 ) elements for the r-Set Packing with t-Overlap and kernels with O(r r k r−t−1 ) and O(m m k m−t−1 ) vertices for the H-Packing with t-Overlap and its edge version, respectively. In all cases, k is the input parameter, while t, r, and m are constants.

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“…In [10], Dell and Marx studied several matching and packing problems, and provided non-trivial lower bounds as well as non-trivial upper bounds for packing some specific graphs such as matchings, P 4 's (here, the packing need not be induced) and K 1,d 's (stars with d leaves). Moser et al [30] studied the problem of packing a fixed connected graph H on vertices in an input graph G (that is, determining whether there exist k vertex disjoint copies of H in G) and designed a kernel with O(k −1 ) vertices. In this context, it is also worth to point out the dichotomy result of Jansen and Marx [25] regarding packing a fixed graph H. Finally, very recently Bessy et al [3] studied FVST where the input tournament is restricted to be a sparse tournament, that is, a tournament where the feedback arc set is a matching.…”

Section: Introductionmentioning

confidence: 99%

Subquadratic Kernels for Implicit 3-Hitting Set and 3-Set Packing Problems

Lokshtanov¹,

Saurabh²,

Thomassé³

et al. 2018

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

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We consider four well-studied NP-complete packing/covering problems on graphs: Feedback Vertex Set in Tournaments (FVST), Cluster Vertex Deletion (CVD), Triangle Packing in Tournaments (TPT) and Induced P 3 -Packing. For these four problems kernels with O(k 2 ) vertices have been known for a long time. 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 sets of size at most 3 (3-Hitting Set). In this paper, we give the first kernels for FVST, CVD, TPT and Induced P 3 -Packing with a subquadratic number of vertices. Specifically, we obtain the following results.• FVST admits a kernel with O(k 3 2 ) vertices.• CVD admits a kernel with O(k 5 3 ) vertices.• TPT admits a kernel with O(k 3 2 ) vertices.• Induced P 3 -Packing admits a kernel with O(k Our results resolve an open problem from WorKer 2010 on the existence of kernels with O(k 2− ) vertices for FVST and CVD. All of our results are based on novel uses of old and new "expansion lemmas", and a weak form of crown decomposition where (i) almost all of the head is used by the solution (as opposed to all), (ii) almost none of the crown is used by the solution (as opposed to none), and (iii) if H is removed from G, then there is almost no interaction between the head and the rest (as opposed to no interaction at all).

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A Problem Kernelization for Graph Packing

Cited by 26 publications

References 27 publications

A kernelization algorithm for d-Hitting Set

A kernelization algorithm for d-Hitting Set

Using Parametric Transformations Toward Polynomial Kernels for Packing Problems Allowing Overlaps

Subquadratic Kernels for Implicit 3-Hitting Set and 3-Set Packing Problems

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