Approximation algorithms for euler genus and related problems

Chandra Chekuri, Anastasios Sidiropoulos

Research output: Contribution to journalArticlepeer-review

Abstract

The Euler genus of a graph is a fundamental and well-studied parameter in graph theory and topology. Computing it has been shown to be NP-hard by Thomassen [J. Algorithms, 10 (1989), pp. 568–576; J. Combin. Theory, Ser. B, 57 (1993), pp. 196–206], and it is known to be fixed-parameter tractable. However, the approximability of the Euler genus is wide open. While the existence of an O(1)-approximation is not ruled out, only an O(n)-approximation [J. Chen, S. P. Kanchi, and A. Kanevsky, Inform. Process. Lett., 61 (1997), pp. 317–322] is known even in bounded-degree graphs. In this paper we give a polynomial-time algorithm which, given a bounded-degree graph of Euler genus g, computes a drawing in a surface of Euler genus gO(1) · logO(1) n. Combined with the upper bound from [J. Chen, S. P. Kanchi, and A. Kanevsky, Inform. Process. Lett., 61 (1997), pp. 317–322], our result also implies a O(n1/2−α)-approximation for some constant α > 0. Using our algorithm for approximating the Euler genus as a subroutine, we obtain, in a uniform fashion, algorithms with approximation ratios of the form OPTO(1) · logO(1) n for several related problems on bounded-degree graphs. These include the problems of orientable genus, crossing number, and planar edge and vertex deletion. Our algorithm and proof of correctness for the crossing number problem are simpler compared to the long and difficult proof in the recent breakthrough by Chuzhoy [Proceedings of the ACM Symposium on Theory of Computing, 2011, pp. 303–312], while essentially obtaining a qualitatively similar result. For planar edge and vertex deletion problems our results are the first to obtain a bound of the form poly(OPT, log n). We also highlight some further applications of our results in the design of algorithms for graphs with small genus. Many such algorithms require that a drawing of the graph is given as part of the input. Our results imply that in several interesting cases, we can implement such algorithms even when the drawing is unknown.

Original languageEnglish (US)
Pages (from-to)1610-1643
Number of pages34
JournalSIAM Journal on Computing
Volume47
Issue number4
DOIs
StatePublished - 2018

Keywords

  • Approximation algorithms
  • Crossing number
  • Euler genus
  • Minimum planarization

ASJC Scopus subject areas

  • General Computer Science
  • General Mathematics

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