Connecting the dots (with minimum crossings)

Akanksha Agrawal, Grzegorz Guśpiel, Jayakrishnan Madathil, Saket Saurabh, Meirav Zehavi

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

2 Scopus citations

Abstract

We study a prototype Crossing Minimization problem, defined as follows. Let F be an infinite family of (possibly vertex-labeled) graphs. Then, given a set P of (possibly labeled) n points in the Euclidean plane, a collection L ⊆ Lines(P) = {ℓ : ℓ is a line segment with both endpoints in P}, and a non-negative integer k, decide if there is a subcollection L0 ⊆ L such that the graph G = (P, L0) is isomorphic to a graph in F and L0 has at most k crossings. By G = (P, L0), we refer to the graph on vertex set P, where two vertices are adjacent if and only if there is a line segment that connects them in L0. Intuitively, in Crossing Minimization, we have a set of locations of interest, and we want to build/draw/exhibit connections between them (where L indicates where it is feasible to have these connections) so that we obtain a structure in F. Natural choices for F are the collections of perfect matchings, Hamiltonian paths, and graphs that contain an (s, t)-path (a path whose endpoints are labeled). While the objective of seeking a solution with few crossings is of interest from a theoretical point of view, it is also well motivated by a wide range of practical considerations. For example, links/roads (such as highways) may be cheaper to build and faster to traverse, and signals/moving objects would collide/interrupt each other less often. Further, graphs with fewer crossings are preferred for graphic user interfaces. As a starting point for a systematic study, we consider a special case of Crossing Minimization. Already for this case, we obtain NP-hardness and W[1]-hardness results, and ETH-based lower bounds. Specifically, suppose that the input also contains a collection D of d non-crossing line segments such that each point in P belongs to exactly one line in D, and L does not contain line segments between points on the same line in D. Clearly, Crossing Minimization is the case where d = n – then, P is in general position. The case of d = 2 is of interest not only because it is the most restricted non-trivial case, but also since it corresponds to a class of graphs that has been well studied – specifically, it is Crossing Minimization where G = (P, L) is a (bipartite) graph with a so called two-layer drawing. For d = 2, we consider three basic choices of F. For perfect matchings, we show (i) NP-hardness with an ETH-based lower bound, (ii) solvability in subexponential parameterized time, and (iii) existence of an O(k2)-vertex kernel. Second, for Hamiltonian paths, we show (i) solvability in subexponential parameterized time, and (ii) existence of an O(k2)-vertex kernel. Lastly, for graphs that contain an (s, t)-path, we show (i) NP-hardness and W[1]-hardness, and (ii) membership in XP.

Original languageEnglish
Title of host publication35th International Symposium on Computational Geometry, SoCG 2019
EditorsGill Barequet, Yusu Wang
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771047
DOIs
StatePublished - 1 Jun 2019
Event35th International Symposium on Computational Geometry, SoCG 2019 - Portland, United States
Duration: 18 Jun 201921 Jun 2019

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume129
ISSN (Print)1868-8969

Conference

Conference35th International Symposium on Computational Geometry, SoCG 2019
Country/TerritoryUnited States
CityPortland
Period18/06/1921/06/19

Keywords

  • Crossing minimization
  • FPT algorithm
  • Parameterized complexity
  • Polynomial kernel
  • W[1]-hardness

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