A tighter insertion-based approximation of the crossing number

Chimani, Markus; Hliněný, Petr

doi:10.1007/s10878-016-0030-z

Cited by 15 publications

(28 citation statements)

References 24 publications

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“…The EIV can be solved in O(n) time using an algorithm by Gutwenger et al [22], which finds a suitable embedding (with the help of SPR-trees) and then executes the EIF-algorithm described above. Now consider the MEIV: Solving it for general k is NP-hard [27], however there exists an O(kn + k 2 )-time approximation algorithm with an additive guarantee of ∆k log k + k 2 [14] that performs well in practice [10]. Put briefly, the EIV-algorithm is run for each of the k edges independently, and a single final embedding is identified by combining the individual (potentially conflicting) solutions via voting.…”

Section: Solving Insertion Problemsmentioning

confidence: 99%

Star-Struck by Fixed Embeddings: Modern Crossing Number Heuristics

Chimani¹,

Ilsen²,

Wiedera³

2021

Preprint

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We present a thorough experimental evaluation of several crossing minimization heuristics that are based on the construction and iterative improvement of a planarization, i.e., a planar representation of a graph with crossings replaced by dummy vertices. The evaluated heuristics include variations and combinations of the well-known planarization method, the recently implemented star reinsertion method, and a new approach proposed herein: the mixed insertion method. Our experiments reveal the importance of several implementation details such as the detection of non-simple crossings (i.e., crossings between adjacent edges or multiple crossings between the same two edges). The most notable finding, however, is that the insertion of stars in a fixed embedding setting is not only significantly faster than the insertion of edges in a variable embedding setting, but also leads to solutions of higher quality.

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Section: Solving Insertion Problemsmentioning

confidence: 99%

Star-Struck by Fixed Embeddings: Modern Crossing Number Heuristics

Chimani¹,

Ilsen²,

Wiedera³

2021

Preprint

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“…Combining the above result with the techniques from [4] we obtain the following algorithm for the crossing number problem (see [12] for details). Corollary 1.3.…”

Section: Introductionmentioning

confidence: 99%

Polylogarithmic approximation for Euler genus on bounded degree graphs

Kawarabayashi

Sidiropoulos

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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Computing the Euler genus of a graph is a fundamental problem in algorithmic graph theory. It has been shown to be NP-hard by [Thomassen '89, Thomassen '97], even for cubic graphs, and a lineartime fixed-parameter algorithm has been obtained by [Mohar '99]. Despite extensive study, the approximability of the Euler genus remains wide open. While the existence of an O(1)-approximation is not ruled out, the currently best-known upper bound is a O(n 1−α)approximation, for some universal constant α > 0 [Kawarabayashi and Sidiropoulos 2017]. We present an O(log 2.5 n)-approximation polynomial time algorithm for this problem on graphs of bounded degree. Prior to our work, the best known result on graphs of bounded degree was a n Ω(1)-approximation [Chekuri and Sidiropoulos 2013]. As an immediate corollary, we also obtain improved approximation algorithms for the crossing number problem and for the minimum vertex planarization problem, on graphs of bounded degree. Specifically, we obtain a polynomial-time O(∆ 2 log 3.5 n)approximation algorithm for the minimum vertex planarization problem, on graphs of maximum degree ∆. Moreover we obtain an algorithm which given a graph of crossing number k, computes a drawing with at most k 2 log O (1) n crossings in polynomial time. This also implies a n 1/2 log O (1) n-approximation polynomial time algorithm. The previously best-known result is a polynomial time algorithm that computes a drawing with k 10 log O (1) crossings, which implies a n 9/10 log O (1) n-approximation algorithm [Chuzhoy 2011]. CCS CONCEPTS • Theory of computation → Graph algorithms analysis; Approximation algorithms analysis.

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“…Specifically, [CMS11] provided an efficient algorithm, that, given an input graph G, and a planarizing set E of k edges for G, draws the graph with O ∆ 3 • k • (OPT cr (G) + k) crossings. Later, Chimani and Hliněnỳ [CH11] improved this bound via a different efficient algorithm to O ∆ • k • (OPT cr (G) + log k) + k 2 . Both works can be viewed as an implementation of the above paradigm.…”

Section: Introductionmentioning

confidence: 99%

“…The bottleneck in using this approach in order to obtain a better than O( √ n)-approximation for Minimum Crossing Number is the bounds of [CMS11] and [CH11], whose algorithms produce a drawing of the graph G with O k • OPT cr (G) + k 2 crossings when ∆ = O(1), where k is the size of the given planarizing set. The quadratic dependence of this bound on k and the linear dependence on k • OPT cr (G) are unacceptable if our goal is to obtain better approximation using this technique.…”

Section: Introductionmentioning

confidence: 99%

“…Unfortunately, Chuzhoy, Madan and Mahabadi [CMM16] have answered this question in the negative; see Section A for details. Their results show that the bounds of [CMS11] and [CH11] are almost tight, and so we cannot hope to break the Θ( √ n) approximation barrier for Minimum Crossing Number using the above paradigm. This seems to doom the only currently known approach for designing approximation algorithms for the problem.…”

Section: Introductionmentioning

confidence: 99%

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Towards Better Approximation of Graph Crossing Number

Chuzhoy

Mahabadi

Tan

2020

Preprint

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Graph Crossing Number is a fundamental and extensively studied problem with wide ranging applications. In this problem, the goal is to draw an input graph G in the plane so as to minimize the number of crossings between the images of its edges. The problem is notoriously difficult, and despite extensive work, non-trivial approximation algorithms are only known for bounded-degree graphs. Even for this special case, the best current algorithm achieves a Õ( √ n)-approximation, while the best current negative results do not rule out constant-factor approximation. All current approximation algorithms for the problem build on the same paradigm, which is also used in practice: compute a set E of edges (called a planarizing set) such that G \ E is planar; compute a planar drawing of G \ E ; then add the drawings of the edges of E to the resulting drawing. Unfortunately, there are examples of graphs G, in which any implementation of this method must incur Ω(OPT 2 ) crossings, where OPT is the value of the optimal solution. This barrier seems to doom the only currently known approach to designing approximation algorithms for the problem, and to prevent it from yielding a better than O( √ n)-approximation. In this paper we propose a new paradigm that allows us to overcome this barrier. We show an algorithm, that, given a bounded-degree graph G and a planarizing set E of its edges, computes another planarizing edge set E with E ⊆ E , such that |E | is relatively small, and there exists a near-optimal drawing of G in which no edges of G \ E participate in crossings. This allows us to reduce the Crossing Number problem to Crossing Number with Rotation System -a variant of the Crossing Number problem, in which the ordering of the edges incident to every vertex is fixed as part of input. In our reduction, we obtain an instance G of this problem, where |E(G )| is roughly bounded by the crossing number of the original graph G. We show a randomized algorithm for this new problem, that allows us to obtain an O(n 1/2− )-approximation for Graph Crossing Number on bounded-degree graphs, for some constant > 0.

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A tighter insertion-based approximation of the crossing number

Cited by 15 publications

References 24 publications

Star-Struck by Fixed Embeddings: Modern Crossing Number Heuristics

Star-Struck by Fixed Embeddings: Modern Crossing Number Heuristics

Polylogarithmic approximation for Euler genus on bounded degree graphs

Towards Better Approximation of Graph Crossing Number

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