Towards Better Approximation of Graph Crossing Number

Chuzhoy, Julia; Mahabadi, Sepideh; Tan, Zihan

doi:10.1109/focs46700.2020.00016

Cited by 6 publications

(20 citation statements)

References 26 publications

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“…Finally, we pose a problem about using MultiInsertion to approximate the tanglegram crossing number, where the bound is modeled after one in [4]. For any tanglegram (T, S, φ), this also requires finding a planar subtanglegram (T I , S φ(I) , φ| I ), and from Corollary 4.18, we know that we do not necessarily want a subtanglegram of maximum size.…”

Section: Elementmentioning

confidence: 99%

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Characterizing Planar Tanglegram Layouts and Applications to Edge Insertion Problems

Liu¹

2022

Preprint

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Tanglegrams are formed by taking two rooted binary trees T and S with the same number of leaves and uniquely matching each leaf in T with a leaf in S. They are usually represented using layouts, which embed the trees and the matching of the leaves into the plane as in Figure 1. Given the numerous ways to construct a layout, one problem of interest is the Tanglegram Layout Problem, which is to efficiently find a layout that minimizes the number of crossings. This parallels a similar problem involving drawings of graphs, where a common approach is to insert edges into a planar subgraph. In this paper, we will explore inserting edges into a planar tanglegram. Previous results on planar tanglegrams include a Kuratowski Theorem, enumeration, and an algorithm for drawing a planar layout [5,16,15]. We start by building on these results and characterizing all planar layouts of a planar tanglegram. We then apply this characterization to construct a quadratic-time algorithm that inserts a single edge optimally. Finally, we generalize some results to multiple edge insertion.

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Section: Elementmentioning

confidence: 99%

“…. , e n } optimally, and current approximation algorithms for graph drawings still use multiple edge insertion with planar subgraphs [4]. Given the role that edge insertion with planar subgraphs plays in graph drawings, it is plausible that edge insertion can play a similar role for tanglegram layouts.…”

Section: Introductionmentioning

confidence: 99%

Characterizing Planar Tanglegram Layouts and Applications to Edge Insertion Problems

Liu¹

2022

Preprint

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“…All of the above results follow the same high-level algorithmic framework, and it was shown by Chuzhoy, Madan and Mahabadi [CMM16] (see [Chu16] for an exposition) that this framework is unlikely to yield a better than O( √ n)-approximation. The most recent result, by Chuzhoy, Mahabadi and Tan [CMT20], obtained an Õ(n 1/2− • poly(∆))-approximation algorithm for some small fixed constant > 0. This result was achieved by proposing a new algorithmic framework for the problem, that departs from the previous approach.…”

Section: Introductionmentioning

confidence: 99%

“…This result was achieved by proposing a new algorithmic framework for the problem, that departs from the previous approach. Specifically, [CMT20] reduced the MCN problem to another problem, called Minimum Crossing Number with Rotation System (MCNwRS) that we discuss below, which appears somewhat easier than the MCN problem, and then provided an algorithm for approximately solving the MCNwRS problem.…”

Section: Introductionmentioning

confidence: 99%

“…Our main result is a randomized O 2 O((log n) 7/8 log log n) • ∆ O(1) -approximation algorithm for MCN. In order to achieve this result, we design a new algorithm for the MCNwRS problem that achieves significantly stronger guarantees than those of [CMT20]. This algorithm, combined with the reduction of [CMT20], immediately implies the improved approximation for the MCN problem.…”

Section: Introductionmentioning

confidence: 99%

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A Subpolynomial Approximation Algorithm for Graph Crossing Number in Low-Degree Graphs

Chuzhoy¹,

Tan²

2022

Preprint

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We consider the classical Minimum Crossing Number problem: given an n-vertex graph G, compute a drawing of G in the plane, while minimizing the number of crossings between the images of its edges. This is a fundamental and extensively studied problem, whose approximability status is widely open. In all currently known approximation algorithms, the approximation factor depends polynomially on ∆ -the maximum vertex degree in G. The best current approximation algorithm achieves an O(n 1/2− • poly(∆ • log n))-approximation, for a small fixed constant , while the best negative result is APX-hardness, leaving a large gap in our understanding of this basic problem. In this paper we design a randomized O 2 O((log n) 7/8 log log n) • poly(∆) -approximation algorithm for Minimum Crossing Number. This is the first approximation algorithm for the problem that achieves a subpolynomial in n approximation factor (albeit only in graphs whose maximum vertex degree is subpolynomial in n).In order to achieve this approximation factor, we design a new algorithm for a closely related problem called Crossing Number with Rotation System, in which, for every vertex v ∈ V (G), the circular ordering, in which the images of the edges incident to v must enter the image of v in the drawing is fixed as part of input. Combining this result with the recent reduction of [Chuzhoy, Mahabadi, Tan '20] immediately yields the improved approximation algorithm for Minimum Crossing Number. We introduce several new technical tools, that we hope will be helpful in obtaining better algorithms for the problem in the future.

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Parameterized analysis and crossing minimization problems

Zehavi

2022

Computer Science Review

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Towards Better Approximation of Graph Crossing Number

Cited by 6 publications

References 26 publications

Characterizing Planar Tanglegram Layouts and Applications to Edge Insertion Problems

Characterizing Planar Tanglegram Layouts and Applications to Edge Insertion Problems

A Subpolynomial Approximation Algorithm for Graph Crossing Number in Low-Degree Graphs

Parameterized analysis and crossing minimization problems

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