Uniform Kernelization Complexity of Hitting Forbidden Minors

Giannopoulou, Archontia C.; Jansen, Bart M. P.; Lokshtanov, Daniel; Saurabh, Saket

doi:10.1145/3029051

Cited by 15 publications

(7 citation statements)

References 38 publications

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“…The size of the Vertex Cover queries generated by the Turing kernelization does not depend on F: the Turing kernelization can be shown to be uniformly polynomial (cf. [20]). However, it remains unknown whether the running time can be made uniformly polynomial, and whether the Turing kernelization can be improved to a traditional kernelization.…”

Section: Resultsmentioning

confidence: 99%

A Turing Kernelization Dichotomy for Structural Parameterizations of $$\mathcal {F}$$ -Minor-Free Deletion

Donkers

Jansen

2019

Graph-Theoretic Concepts in Computer Science

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For a fixed finite family of graphs F, the F-Minor-Free Deletion problem takes as input a graph G and an integer and asks whether there exists a set X ⊆ V (G) of size at most such that G − X is F-minor-free. For F = {K 2 } and F = {K 3 } this encodes Vertex Cover and Feedback Vertex Set respectively. When parameterized by the feedback vertex number of G these two problems are known to admit a polynomial kernelization. Such a polynomial kernelization also exists for any F containing a planar graph but no forests. In this paper we show that F-Minor-Free Deletion parameterized by the feedback vertex number is MK[2]-hard for F = {P 3 }. This rules out the existence of a polynomial kernel assuming NP ⊆ coNP/poly, and also gives evidence that the problem does not admit a polynomial Turing kernel. Our hardness result generalizes to any F not containing a P 3 -subgraph-free graph, using as parameter the vertex-deletion distance to treewidth min tw(F), where min tw(F) denotes the minimum treewidth of the graphs in F. For the other case, where F contains a P 3 -subgraph-free graph, we present a polynomial Turing kernelization. Our results extend to F-Subgraph-Free Deletion. ACM Subject ClassificationTheory of computation → Graph algorithms analysis, Theory of computation → Parameterized complexity and exact algorithms

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Section: Resultsmentioning

confidence: 99%

A Turing Kernelization Dichotomy for Structural Parameterizations of $$\mathcal {F}$$ -Minor-Free Deletion

Donkers

Jansen

2019

Graph-Theoretic Concepts in Computer Science

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“…We refer (in particular) to Dell and Marx [2012] and Hermelin and Wu [2012] for such results for disjoint packings in graphs and hypergraphs. Recently, Giannopoulou et al introduced the notion of uniform kernelization [Giannopoulou et al 2014] for problem families similar to what we considered. This basically raises the question of whether polynomial kernel sizes could be proven such that the exponent of the kernel bound does not depend (in our case) on r or on t. Notice that for t-Membership, we somehow came half of the way, as we have shown kernel sizes that are uniform with respect to t. In particular, {K 3 }-Packing with t-Membership has a uniform kernelization.…”

Section: Discussionmentioning

confidence: 97%

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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“…These problems have led to identification of several new techniques and ideas in the field. Recent years have seen a plethora of results around vertex and edge deletion problems, in the domain of parameterized complexity [3,4,[8][9][10][11][12]. In this paper, we continue this line of research and study two vertex deletion problems.…”

Section: Introductionmentioning

confidence: 92%

Quick but Odd Growth of Cacti

et al. 2017

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Let F be a family of graphs. Given an input graph G and a positive integer k, testing whether G has a k-sized subset of vertices S, such that G \ S belongs to F, is a prototype vertex deletion problem. These type of problems have attracted a lot of attention in recent times in the domain of parameterized complexity. In this paper, we study two such problems; when F is either a family of cactus graphs or a family of odd-cactus graphs. A graph H is called a cactus graph if every pair of cycles in H intersect on at most one vertex. Furthermore, a cactus graph H is called an odd cactus, if every cycle of H is of odd length. Let us denote by C and C odd , families of cactus and odd cactus, respectively. The vertex deletion problems corresponding to C and C odd are called Diamond Hitting Set and Even Cycle Transversal, respectively. In this paper we design randomized algorithms with running time 12 k n O(1) for both these problems. Our algorithms considerably improve the running time for Diamond Hitting Set and Even Cycle Transversal, compared to what is known about them.

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Uniform Kernelization Complexity of Hitting Forbidden Minors

Cited by 15 publications

References 38 publications

A Turing Kernelization Dichotomy for Structural Parameterizations of $$\mathcal {F}$$ -Minor-Free Deletion

A Turing Kernelization Dichotomy for Structural Parameterizations of $$\mathcal {F}$$ -Minor-Free Deletion

Using Parametric Transformations Toward Polynomial Kernels for Packing Problems Allowing Overlaps

Quick but Odd Growth of Cacti

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