Grid Recognition: Classical and Parameterized Computational Perspectives

Siddharth Gupta, Guy Sa’ar, Meirav Zehavi

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

2 Scopus citations

Abstract

Grid graphs, and, more generally, k × r grid graphs, form one of the most basic classes of geometric graphs. Over the past few decades, a large body of works studied the (in)tractability of various computational problems on grid graphs, which often yield substantially faster algorithms than general graphs. Unfortunately, the recognition of a grid graph (given a graph G, decide whether it can be embedded into a grid graph) is particularly hard – it was shown to be NP-hard even on trees of pathwidth 3 already in 1987. Yet, in this paper, we provide several positive results in this regard in the framework of parameterized complexity (additionally, we present new and complementary hardness results). Specifically, our contribution is threefold. First, we show that the problem is fixed-parameter tractable (FPT) parameterized by k + mcc where mcc is the maximum size of a connected component of G. This also implies that the problem is FPT parameterized by td + k where td is the treedepth of G, as td ≤ mcc (to be compared with the hardness for pathwidth 2 where k = 3). (We note that when k and r are unrestricted, the problem is trivially FPT parameterized by td.) Further, we derive as a corollary that strip packing is FPT with respect to the height of the strip plus the maximum of the dimensions of the packed rectangles, which was previously only known to be in XP. Second, we present a new parameterization, denoted aG, relating graph distance to geometric distance, which may be of independent interest. We show that the problem is para-NP-hard parameterized by aG, but FPT parameterized by aG on trees, as well as FPT parameterized by k + aG. Third, we show that the recognition of k × r grid graphs is NP-hard on graphs of pathwidth 2 where k = 3. Moreover, when k and r are unrestricted, we show that the problem is NP-hard on trees of pathwidth 2, but trivially solvable in polynomial time on graphs of pathwidth 1.

Original languageEnglish
Title of host publication32nd International Symposium on Algorithms and Computation, ISAAC 2021
EditorsHee-Kap Ahn, Kunihiko Sadakane
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772143
DOIs
StatePublished - 1 Dec 2021
Event32nd International Symposium on Algorithms and Computation, ISAAC 2021 - Fukuoka, Japan
Duration: 6 Dec 20218 Dec 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume212
ISSN (Print)1868-8969

Conference

Conference32nd International Symposium on Algorithms and Computation, ISAAC 2021
Country/TerritoryJapan
CityFukuoka
Period6/12/218/12/21

Keywords

  • Grid graph
  • Grid recognition
  • Parameterized complexity

ASJC Scopus subject areas

  • Software

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