Subexponential Parameterized Algorithms on Disk Graphs (Extended Abstract)

Abstract

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Subexponential Parameterized Algorithms on Disk Graphs (Extended Abstract)Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Jie Xue, and Meirav ZehaviDaniel Lokshtanov, Fahad Panolan, Saket Saurabh, Jie Xue, and Meirav Zehavipp.2005 - 2031Chapter DOI:https://doi.org/10.1137/1.9781611977073.80PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract One of the most celebrated results in Parameterized Complexity is the Bidimensionality theory of Demaine et al. [J. ACM, 2005], which has yielded, over the past two decades, numerous subexponential-time fixed-parameter tractable (FPT) algorithms for various problems on planar (and H-minor-free) graphs. At the heart of this theory is the proof of sublinear bounds in terms of solution size on the treewidth of a given graph. Inspired by this theory, in recent years, significant efforts have been devoted to design subexponential-time FPT algorithms for problems on geometric graph classes that utilize new treewidth bounds, in particular (but not only) for unit disk graphs [Fomin et al., SODA'12; Fomin et al., DCG'19; Panolan et al., SODA'19; Fomin et al. SoCG'20]. In this paper, we aim to attain such results on disk graphs, a broad class of graphs that generalizes both the classes of planar graphs and unit disk graphs, and thereby unify the aforementioned research frontiers for planar and unit disk graphs. Our main contribution is an approach to design subexponential-time FPT algorithms for problems on disk graphs, which we apply to several well-studied graph problems. At the heart of our approach lie two new combinatorial theorems concerning the treewidth of disk graphs having a realization of bounded ply (or maximum clique size) that are of independent interest. In particular, we prove a stronger version of the following treewidth bound: Let G be a disk graph that has some realization of ply p and no false twins, and M ⊆ V(G) such that G has no triangle with exactly one vertex from M, and G–M has treewidth w. Then, the treewidth of G is . Among our applications are the first subexponential-time FPT algorithms for several problems on disk graphs, including Triangle Hitting, Feedback Vertex Set and Odd Cycle Transversal (OCT). Previously, subexponential-time FPT algorithms for these problems were only known on planar graphs and unit disk graphs (excluding OCT, which was only known to admit such an algorithm on planar graphs). Our algorithms are robust, in particular, they do not require a geometric realization of the input graph (for all aforementioned problems), and they generalize to the weighted and counting versions of all aforementioned problems except for OCT. Previous chapter Next chapter RelatedDetails Published:2022eISBN:978-1-61197-707-3 https://doi.org/10.1137/1.9781611977073Book Series Name:ProceedingsBook Code:PRDA22Book Pages:xvii + 3771