An improved kernelization algorithm for r-Set Packing

Abu-Khzam, Faisal N.

doi:10.1016/j.ipl.2010.04.020

Cited by 54 publications

(84 citation statements)

References 13 publications

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“…Recall that a parameterized problem is fixed-parameter tractable (FPT) with a given parameterization if there is an algorithm with running time f (k) · n O (1) , where n is the size of the instance, k is the value of the parameter associated with the instance, and f is an arbitrary computable function depending only on the parameter k (see the monographs [44,50,87] or the survey [41] in this volume for more background). That is, the problem can be solved in polynomial time for every fixed value of the parameter k and the exponent does not depend on the parameter.…”

Section: The Optimality Programmentioning

confidence: 99%

“…We sketch how such lower bounds can be proved using a complexitytheoretic assumption called Exponential Time Hypothesis (ETH). (An alternate way to discuss these results is via the complexity class M [1] and for some of the results even the weaker FPT = W [1] hypothesis is sufficient. However, to keep our discussion focused, we describe only results based on ETH here.)…”

Section: The Optimality Programmentioning

confidence: 99%

“…w · n O (1) . However, there are still a number of problems where the best running time seems to be "stuck" at 2 O(k log k) · n O (1) .…”

Section: Corollary 24 Assuming Eth There Is Nomentioning

confidence: 99%

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What’s Next? Future Directions in Parameterized Complexity

Marx

2012

The Multivariate Algorithmic Revolution and Beyond

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Abstract. The progress in parameterized complexity has been very significant in recent years, with new research questions and directions, such as kernelization lower bounds, appearing and receiving considerable attention. This speculative article tries to identify new directions that might become similar hot topics in the future. First, we point out that the search for optimality in parameterized complexity already has good foundations, but lots of interesting work can be still done in this area. The systematic study of kernelization became a very successful research direction in recent years. We look at what general conclusions one can draw from these results and we argue that the systematic study of other algorithmic techniques should be modeled after the study of kernelization. In particular, we set up a framework for understanding which problems can be solved by branching algorithms. Finally, we discuss that the domain of directed graph problems is a challenging area which can potentially see significant progress in the following years.

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Section: The Optimality Programmentioning

confidence: 99%

Section: The Optimality Programmentioning

confidence: 99%

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What’s Next? Future Directions in Parameterized Complexity

Marx

2012

The Multivariate Algorithmic Revolution and Beyond

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“…1 In this problem, the input consists of a universe U , a family F containing sets of size at most d over U , and in integer k. The objective is to determine whether there exists a set S ⊆ U of size at most k that intersects every set in F. Abu-Khzam [2] gave an improved kernel for d-Hitting Set, still with O(k d ) sets, but with O(k d−1 ) elements. The importance of the d-Hitting Set problem stems from the number of other problems that can be re-cast in terms of it.…”

Section: Introductionmentioning

confidence: 99%

“…• CVD admits a kernel with O(k The Sunflower Lemma based kernel for d-Hitting Set and the improvement of Abu-Khzam [2] can also be applied to the d-Set Packing problem [1]. Here, the input consists of a universe U and a family F of sets of size d over U , together with an integer k. The task is to determine whether there exists a subfamily F of k pairwise disjoint sets.…”

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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Kernelization of Arc Disjoint Cycle Packing in $$\alpha $$-Bounded Digraphs

Sahu

Saurabh

2020

Computer Science – Theory and Applications

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An improved kernelization algorithm for r-Set Packing

Cited by 54 publications

References 13 publications

What’s Next? Future Directions in Parameterized Complexity

What’s Next? Future Directions in Parameterized Complexity

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

Kernelization of Arc Disjoint Cycle Packing in $$\alpha $$-Bounded Digraphs

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