Feedback Vertex Set Inspired Kernel for Chordal Vertex Deletion

Agrawal, Akanksha; Lokshtanov, Daniel; Misra, Pranabendu; Saurabh, Saket; Zehavi, Meirav

doi:10.1145/3284356

Cited by 13 publications

(21 citation statements)

References 48 publications

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“…Exponential Time Hypothesis states that 3 > 3 . In particular, ETH implies that k -SatiSFiaBility cannot be solved in time 2 o(n) n O (1) .…”

Section: Proposition 2 Every N-vertex Chordal Graph G Admits a Nice Tree Decomposition T = (T {X T } T∈v(t)mentioning

confidence: 99%

“…for every i ∈ {1, … , n} . Because a d-coloring ordering can be found in polynomial (in fact, linear) time [51], the total running time is 2 1) . ◻…”

Section: Subexponential Algorithms For Induced D-colorable Subgraphsmentioning

confidence: 99%

“…Since an algorithm for independent Set of running time 2 o(n) will refute the Exponential Time Hypothesis (ETH) of Impagliazzo, Paturi and Zane [39,40], and because k ≤ n 2 , the existence of an algorithm of running time 1) for some > 0 (which is polynomial for k = (log n) 2∕(1−2 ) ) is unlikely. Interestingly, as we shall see, independent Set (and many other problems) are solvable in time 2 k 1∕2 log k ⋅ n O (1) . Leizhen Cai in [12] introduced a convenient notation for "perturbed" graph classes.…”

Section: Introductionmentioning

confidence: 99%

“…MiniMuM Fill-in (under the name Chordal graph CoMpletion) was one of the 12 open problems presented at the end of the first edition of Garey and Johnson's book [31] and it was proved to be NP-complete by Yannakakis [64]. Kaplan et al [44] proved that MiniMuM Fill-in is fixed parameter tractable by giving an algorithm of running time 16 k ⋅ n O (1) . There was a chain of algorithmic improvements resulting in decreasing the constant in the base of the exponents [7,11,45] resulting with a subexponential algorithm of running time…”

Section: Introductionmentioning

confidence: 99%

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Subexponential Parameterized Algorithms and Kernelization on Almost Chordal Graphs

Fomin

Golovach

2021

Algorithmica

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We study algorithmic properties of the graph class $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e , that is, graphs that can be turned into a chordal graph by adding at most k edges or, equivalently, the class of graphs of fill-in at most k. It appears that a number of fundamental intractable optimization problems being parameterized by k admit subexponential algorithms on graphs from $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e . More precisely, we identify a large class of optimization problems on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e solvable in time $$2^{{\mathcal{O}}(\sqrt{k}\log k)}\cdot n^{{\mathcal{O}}(1)}$$ 2 O ( k log k ) · n O ( 1 ) . Examples of the problems from this class are finding an independent set of maximum weight, finding a feedback vertex set or an odd cycle transversal of minimum weight, or the problem of finding a maximum induced planar subgraph. On the other hand, we show that for some fundamental optimization problems, like finding an optimal graph coloring or finding a maximum clique, are FPT on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e when parameterized by k but do not admit subexponential in k algorithms unless ETH fails. Besides subexponential time algorithms, the class of $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e graphs appears to be appealing from the perspective of kernelization (with parameter k). While it is possible to show that most of the weighted variants of optimization problems do not admit polynomial in k kernels on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e graphs, this does not exclude the existence of Turing kernelization and kernelization for unweighted graphs. In particular, we construct a polynomial Turing kernel for Weighted Clique on $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e graphs. For (unweighted) Independent Set we design polynomial kernels on two interesting subclasses of $${\textsc {Chordal}}{-ke}$$ C H O R D A L - k e , namely, $${\textsc {Interval}}{-ke}$$ I N T E R V A L - k e and $${\textsc {Split}}{-ke}$$ S P L I T - k e graphs.

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“…Exponential Time Hypothesis states that 3 > 3 . In particular, ETH implies that k -SatiSFiaBility cannot be solved in time 2 o(n) n O (1) .…”

Section: Proposition 2 Every N-vertex Chordal Graph G Admits a Nice Tree Decomposition T = (T {X T } T∈v(t)mentioning

confidence: 99%

“…for every i ∈ {1, … , n} . Because a d-coloring ordering can be found in polynomial (in fact, linear) time [51], the total running time is 2 1) . ◻…”

Section: Subexponential Algorithms For Induced D-colorable Subgraphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Subexponential Parameterized Algorithms and Kernelization on Almost Chordal Graphs

Fomin

Golovach

2021

Algorithmica

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“…Chordal Vertex Deletion [ALM + 18, JP17, KK18] and Odd Cycle Transversal [ACMM05] are other primary examples of H-Vertex Deletion; they can be captured as a subgraph deletion when F is the set of all chordless or odd cycles. The problem of reducing other width parameters (e.g., rankwidth, cliquewidth) have been studied [ALM + 18].…”

Section: Related Workmentioning

confidence: 99%

Losing Treewidth by Separating Subsets

Gupta¹,

Lee²,

Li³

et al. 2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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We study the problem of deleting the smallest set S of vertices (resp. edges) from a given graph G such that the induced subgraph (resp. subgraph) G \ S belongs to some class H. We consider the case where graphs in H have treewidth bounded by t, and give a general framework to obtain approximation algorithms for both vertex and edge-deletion settings from approximation algorithms for certain natural graph partitioning problems called k-Subset Vertex Separator and k-Subset Edge Separator, respectively.For the vertex deletion setting, our framework combined with the current best result for k-Subset Vertex Separator, improves approximation ratios for basic problems such as k-Treewidth Vertex Deletion and Planar-F Vertex Deletion. Our algorithms are simpler than previous works and give the first deterministic and uniform approximation algorithms under the natural parameterization.For the edge deletion setting, we give improved approximation algorithms for k-Subset Edge Separator combining ideas from LP relaxations and important separators. We present their applications in bounded-degree graphs, and also give an APX-hardness result for the edge deletion problems.3 The conference version of [Lee18] presents a randomized algorithm, but the journal version derandomized it.

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A Polynomial Kernel for Distance-Hereditary Vertex Deletion

Kim

Kwon

2021

Algorithmica

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Feedback Vertex Set Inspired Kernel for Chordal Vertex Deletion

Cited by 13 publications

References 48 publications

Subexponential Parameterized Algorithms and Kernelization on Almost Chordal Graphs

Subexponential Parameterized Algorithms and Kernelization on Almost Chordal Graphs

Losing Treewidth by Separating Subsets

A Polynomial Kernel for Distance-Hereditary Vertex Deletion

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