Contraction Decomposition in Unit Disk Graphs and Algorithmic Applications in Parameterized Complexity

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 an “almost” 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 though under a relaxation of the original formulation.

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 \(\mathcal {O}(k) \) tree decompositions of G with the following properties. Each bag in any of these tree decompositions can be partitioned into \(\mathcal {O}(k) \) 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.