Losing Treewidth by Separating Subsets

Gupta, Anupam; Lee, Euiwoong; Li, Jason; Manurangsi, Pasin; Włodarczyk, Michał

doi:10.1137/1.9781611975482.104

Cited by 22 publications

(28 citation statements)

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“…3 However, it is not immediately clear if their approach can be extended to WPF-MFD. 4 Very recently, Gupta et al [22] have given O(log ) approximation algorithm for (unweighted) Planar F-Minor-Free Deletion, where is the maximum number of vertices in any planar graph in F.…”

Section: :6mentioning

confidence: 99%

Polylogarithmic Approximation Algorithms for Weighted-ℱ-deletion Problems

Agrawal

Lokshtanov

Misra

et al. 2020

ACM Trans. Algorithms

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Let F be a family of graphs. A canonical vertex deletion problem corresponding to F is defined as follows: given an n-vertex undirected graph G and a weight function w : V (G) → R + , find a minimum weight subset S ⊆ V (G) such that G−S belongs to F. This is known as Weighted F Vertex Deletion problem. In this paper we devise a recursive scheme to obtain O(log O(1) n)approximation algorithms for such problems, building upon the classical technique of finding balanced separators in a graph. Roughly speaking, our scheme applies to those problems, where an optimum solution S together with a well-structured set X, form a balanced separator of the input graph. In this paper, we obtain the first O(log O(1) n)-approximation algorithms for the following vertex deletion problems. Let F be a finite set of graphs containing a planar graph, and F = G(F) be the family of graphs such that every graph H ∈ G(F) excludes all graphs in F as minors. The vertex deletion problem corresponding to F = G(F) is the Weighted Planar F-Minor-Free Deletion (WPF-MFD) problem. We give randomized and deterministic approximation algorithms for WPF-MFD with ratios O(log 1.5 n) and O(log 2 n), respectively. Previously, only a randomized constant factor approximation algorithm for the unweighted version of the problem was known [FOCS 2012]. We give an O(log 2 n)-factor approximation algorithm for Weighted Chordal Vertex Deletion (WCVD), the vertex deletion problem to the family of chordal graphs. On the way to this algorithm, we also obtain a constant factor approximation algorithm for Multicut on chordal graphs. We give an O(log 3 n)-factor approximation algorithm for Weighted Distance Hereditary Vertex Deletion (WDHVD), also known as Weighted Rankwidth-1 Vertex Deletion (WR-1VD). This is the vertex deletion problem to the family of distance hereditary graphs, or equivalently, the family of graphs of rankwidth one. We believe that our recursive scheme can be applied to obtain O(log O(1) n)-approximation algorithms for many other problems as well.

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Section: :6mentioning

confidence: 99%

Polylogarithmic Approximation Algorithms for Weighted-ℱ-deletion Problems

Agrawal

Lokshtanov

Misra

et al. 2020

ACM Trans. Algorithms

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“…Fomin et al [247] gave a randomized f (k)-approximation algorithm that runs in g(k) • nm for some computable functions f and g. The approximation ratio was improved by Gupta et al [255] that gave a deterministic O(log k)-approximation algorithm that runs in f (k)…”

Section: Treewidth and Planar Minor Deletionmentioning

confidence: 99%

“…Therefore, in order to solve H-MINOR DELETION, one can first solve TREEWIDTH k-DELETION to reduce the treewidth to k and then solve H-MINOR DELETION optimally using Courcelle's theorem [250]. Combined with the above algorithm for TREEWIDTH k-DELETION [255], this strategy yields an O(log k)-approximation algorithm that runs in f (|H|) • n O(1) time.…”

Section: Treewidth and Planar Minor Deletionmentioning

confidence: 99%

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A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

et al. 2020

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“…In fact, for every planar graph H, there is even a constant factor approximation algorithm for computing τ H (G). Indeed, a randomized constant factor approximation was first developed by Fomin, Lokshtanov, Misra, and Saurabh [21], and very recently a deterministic one was obtained by Gupta, Lee, Li, Manurangsi, and Włodarczyk [27].…”

Section: Approximation Algorithms For Packing and Covering Modelsmentioning

confidence: 99%

A tight Erdős-Pósa function for planar minors

Batenburg¹,

Huynh²,

Joret³

et al. 2019

AiC

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Let F be a family of graphs. Then for every graph G the maximum number of disjoint subgraphs of G, each isomorphic to a member of F, is at most the minimum size of a set of vertices that intersects every subgraph of G isomorphic to a member of F. We say that F packs if equality holds for every graph G. Only very few families pack. As the next best weakening we say that F has the Erdős-Pósa property if there exists a function f such that for every graph G and integer k>0 the graph G has either k disjoint subgraphs each isomorphic to a member of F or a set of at most f(k) vertices that intersects every subgraph of G isomorphic to a member of F. The name is motivated by a classical 1965 result of Erdős and Pósa stating that for every graph G and integer k>0 the graph G has either k disjoint cycles or a set of O(klogk) vertices that intersects every cycle. Thus the family of all cycles has the Erdős-Pósa property with f(k)=O(klogk). In contrast, the family of odd cycles fails to have the Erdős-Pósa property. For every integer ℓ, a sufficiently large Escher Wall has an embedding in the projective plane such that every face is even and every homotopically non-trivial closed curve intersects the graph at least ℓ times. In particular, it contains no set of ℓ vertices such that each odd cycle contains at least one them, yet it has no two disjoint odd cycles. By now there is a large body of literature proving that various families F have the Erdős-Pósa property. A very general theorem of Robertson and Seymour says that for every planar graph H the family F(H) of all graphs with a minor isomorphic to H has the Erdős-Pósa property. (When H is non-planar, F(H) does not have the Erdős-Pósa property.) The present paper proves that for every planar graph H the family F(H) has the Erdős-Pósa property with f(k)=O(klogk), which is asymptotically best possible for every graph H with at least one cycle.

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Losing Treewidth by Separating Subsets

Cited by 22 publications

References 50 publications

Polylogarithmic Approximation Algorithms for Weighted-ℱ-deletion Problems

Polylogarithmic Approximation Algorithms for Weighted-ℱ-deletion Problems

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

A tight Erdős-Pósa function for planar minors

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