TY - GEN
T1 - A Framework for Approximation Schemes on Disk Graphs
AU - Lokshtanov, Daniel
AU - Panolan, Fahad
AU - Saurabh, Saket
AU - Xue, Jie
AU - Zehavi, Meirav
N1 - Funding Information:
∗The full version of the paper can be accessed at https://arxiv.org/abs/2211.02717 †University of California, Santa Barbara, USA. Email: [email protected]. Supported by NSF award CCF-2008838. ‡IIT Hyderabad, India. Email: [email protected]. §Institute of Mathematical Sciences, Chennai, India. Email: [email protected]. Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 819416), and the author also acknowledges the support of Swarnajayanti Fellowship grant DST/SJF/MSA-01/2017-18. ¶New York University Shanghai, China. Email: [email protected]. ‖Ben-Gurion University, Israel. Email: [email protected]. Supported by the European Research Council (ERC) grant titled PARAPATH.
Funding Information:
Supported by NSF award CCF-2008838. Supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 819416), and the author also acknowledges the support of Swarnajayanti Fellowship grant DST/SJF/MSA-01/2017-18. Supported by the European Research Council (ERC) grant titled PARAPATH.
Publisher Copyright:
Copyright © 2023 by SIAM.
PY - 2023/1/1
Y1 - 2023/1/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85167775263&partnerID=8YFLogxK
U2 - 10.1137/1.9781611977554.ch84
DO - 10.1137/1.9781611977554.ch84
M3 - Conference contribution
AN - SCOPUS:85167775263
T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
SP - 2228
EP - 2241
BT - 34th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2023
PB - Association for Computing Machinery
T2 - 34th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2023
Y2 - 22 January 2023 through 25 January 2023
ER -