A Framework for Approximation Schemes on Disk Graphs

Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Jie Xue, Meirav Zehavi

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Scopus citations

Abstract

We initiate a systematic study of approximation schemes for fundamental optimization problems on disk graphs, a common generalization of both planar graphs and unit-disk graphs. Our main contribution is a general framework for designing efficient polynomial-time approximation schemes (EPTASes) for vertex-deletion problems on disk graphs, which results in EPTASes for many fundamental problems including VERTEX COVER, FEEDBACK VERTEX SET, SMALL CYCLE HITTING (in particular, TRIANGLE HITTING), Pk-VERTEX DELETION for k ∈ {3, 4, 5}, PATH DELETION, PATHWIDTH 1-DELETION, COMPONENT ORDER CONNECTIVITY, BOUNDED DEGREE DELETION, PSEUDOFOREST DELETION, FINITE-TYPE COMPONENT DELETION, etc. All EPTASes obtained using our framework are robust in the sense that they do not require a realization of the input disk graph (in fact, we allow the input to be any graph, and our algorithms either output a correct approximation solution for the problem or conclude that the input graph is not a disk graph). To the best of our knowledge, prior to this work, the only problems known to admit PTASes or EPTASes on disk graphs are MAXIMUM CLIQUE, INDEPENDENT SET, DOMINATING SET, and VERTEX COVER, among which the existing PTAS [Erlebach et al., SICOMP'05] and EPTAS [Leeuwen, SWAT'06] for VERTEX COVER require a realization of the input disk graph (while ours does not). The core of our framework is a reduction for a broad class of (approximation) vertex-deletion problems from (general) disk graphs to disk graphs of bounded local radius, which is a new invariant of disk graphs introduced in this work. Disk graphs of bounded local radius can be viewed as a “mild” generalization of planar graphs, which preserves certain nice properties of planar graphs. Specifically, we prove that disk graphs of bounded local radius admit the Excluded Grid Minor property and have locally bounded treewidth. This allows existing techniques for designing approximation schemes on planar graphs (e.g., bidimensionality and Baker's technique) to be directly applied to disk graphs of bounded local radius.

Original languageEnglish
Title of host publication34th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2023
PublisherAssociation for Computing Machinery
Pages2228-2241
Number of pages14
ISBN (Electronic)9781611977554
DOIs
StatePublished - 1 Jan 2023
Event34th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2023 - Florence, Italy
Duration: 22 Jan 202325 Jan 2023

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
Volume2023-January

Conference

Conference34th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2023
Country/TerritoryItaly
CityFlorence
Period22/01/2325/01/23

ASJC Scopus subject areas

  • Software
  • Mathematics (all)

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