Contraction decomposition in unit disk graphs and algorithmic applications in parameterized complexity

Fahad Panolan, Saket Saurabh, Meirav Zehavi

Research output: Contribution to conferencePaperpeer-review

8 Scopus citations

Abstract

We give a new decomposition theorem in unit disk graphs (UDGs) and demonstrate its applicability in the fields of Structural Graph Theory and Parameterized Complexity. First, our new decomposition theorem shows that the class of UDGs admits a Contraction Decomposition Theorem. Prior studies on this topic exhibited that the classes of planar graphs [Klein, SICOMP, 2008], graphs of bounded genus [Demaine, Hajiaghayi and Mohar, Combinatorica 2010] and H-minor free graphs [Demaine, Hajiaghayi and Kawarabayashi, STOC 2011] admit a Contraction Decomposition Theorem. Even bounded-degree UDGs can contain arbitrarily large cliques as minors, therefore our result is a significant advance in the study of contraction decompositions. Additionally, this result answers an open question posed by Hajiaghayi (www.youtube.com/watch?v=2Bq2gy1N01w) regarding the existence of contraction decompositions for classes of graphs beyond H-minor free graphs. Second, we present a “parameteric version” of our new decomposition theorem. We prove that there is an algorithm that given a UDG G and a positive integer k, runs in polynomial time and outputs a collection of O(k) tree decompositions of G with the following properties. Each bag in any of these tree decompositions can be partitioned into O(k) connected pieces (we call this measure the chunkiness of the tree decomposition). Moreover, for any subset S of at most k edges in G, there is a tree decomposition in the collection such that S is well preserved in the decomposition in the following sense. For any bag in the tree decomposition and any edge in S with both endpoints in the bag, either its endpoints lie in different pieces or they lie in a piece which is a clique. Having this decomposition at hand, we show that the design of parameterized algorithms for some cut problems becomes elementary. In particular, our algorithmic applications include single-exponential (or slightly super-exponential) algorithms for well-studied problems such as Min Bisection, Steiner Cut, s-Way Cut, and Edge Multiway Cut-Uncut on UDGs; these algorithms are substantially faster than the best known algorithms for these problems on general graphs.

Original languageEnglish
Pages1035-1054
Number of pages20
DOIs
StatePublished - 1 Jan 2019
Event30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019 - San Diego, United States
Duration: 6 Jan 20199 Jan 2019

Conference

Conference30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019
Country/TerritoryUnited States
CitySan Diego
Period6/01/199/01/19

ASJC Scopus subject areas

  • Software
  • Mathematics (all)

Fingerprint

Dive into the research topics of 'Contraction decomposition in unit disk graphs and algorithmic applications in parameterized complexity'. Together they form a unique fingerprint.

Cite this