Approximation algorithms for Euler genus and related problems

Chandra Chekuri, Anastasios Sidiropoulos

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 [23], [24], 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 [3] is known even in bounded degree graphs. In this paper we give a polynomialtime algorithm which on input a bounded-degree graph of Euler genus g, computes a drawing into a surface of Euler genus gO(1).logO(1) n. Combined with the upper bound from [3], 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 unified 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 problems. Our algorithm and proof of correctness for the crossing number problem is simpler compared to the long and difficult proof in the recent breakthrough by Chuzhoy [5], while essentially obtaining a qualitatively similar result. For planar edge and vertex deletion problems our results are the first to obtain a bound of 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)
Title of host publicationProceedings - 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013
Pages167-176
Number of pages10
DOIs
StatePublished - Dec 1 2013
Event2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013 - Berkeley, CA, United States
Duration: Oct 27 2013Oct 29 2013

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428

Other

Other2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013
Country/TerritoryUnited States
CityBerkeley, CA
Period10/27/1310/29/13

ASJC Scopus subject areas

  • Computer Science(all)

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