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Recognition and Proper Coloring of Unit Segment Intersection Graphs

Authors Robert D. Barish , Tetsuo Shibuya

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Author Details

Robert D. Barish
  • Division of Medical Data Informatics, Human Genome Center, Institute of Medical Science, University of Tokyo, Japan
Tetsuo Shibuya
  • Division of Medical Data Informatics, Human Genome Center, Institute of Medical Science, University of Tokyo, Japan

Funding

  • Shibuya, Tetsuo: This work was supported by JSPS Kakenhi grants {23H03345, 23K18501, 20H05967, 21H05052}.

Cite AsGet BibTex

Robert D. Barish and Tetsuo Shibuya. Recognition and Proper Coloring of Unit Segment Intersection Graphs. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 5:1-5:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SWAT.2024.5

Abstract

In this work, we concern ourselves with the fine-grained complexity of recognition and proper coloring problems on highly restricted classes of geometric intersection graphs of "thin" objects (i.e., objects with unbounded aspect ratios). As a point of motivation, we remark that there has been significant interest in finding algorithmic lower bounds for classic decision and optimization problems on these types of graphs, as they appear to escape the net of known planar or geometric separator theorems for "fat" objects (i.e., objects with bounded aspect ratios). In particular, letting n be the order of a geometric intersection graph, and assuming a geometric ply bound, per what is known as the "square root phenomenon", these separator theorems often imply the existence of 𝒪(2^√n) algorithms for problems ranging from finding proper colorings to finding Hamiltonian cycles. However, in contrast, it is known for instance that no 2^o(n) time algorithm can exist under the Exponential Time Hypothesis (ETH) for proper 6-coloring intersection graphs of line segments embedded in the plane (Biró et. al.; J. Comput. Geom. 9(2); pp. 47-80; 2018). We begin by establishing algorithmic lower bounds for proper k-coloring and recognition problems of intersection graphs of line segments embedded in the plane under the most stringent constraints possible that allow either problem to be non-trivial. In particular, we consider the class UNIT-PURE-k-DIR of unit segment geometric intersection graphs, in which segments are constrained to lie in at most k directions in the plane, and no two parallel segments are permitted to intersect. Here, under the ETH, we show for every k ≥ 3 that no 2^o(√{n/k}) time algorithm can exist for either recognizing or proper k-coloring UNIT-PURE-k-DIR graphs of order n. In addition, for every k ≥ 4, we establish the same algorithmic lower bound under the ETH for the problem of proper (k-1)-coloring UNIT-PURE-k-DIR graphs when provided a list of segment coordinates specified using 𝒪(n ⋅ k) bits witnessing graph class membership. As a consequence of our approach, we are also able to show that the problem of properly 3-coloring an arbitrary graph on m edges can be reduced in 𝒪(m²) time to the problem of properly (k-1)-coloring a UNIT-PURE-k-DIR graph. Finally, we consider a slightly less constrained class of geometric intersection graphs of lines (of unbounded length) in which line-line intersections must occur on any one of (r = 3) parallel planes in ℝ³. In this context, for every k ≥ 3, we show that no 2^o(n/k) time algorithm can exist for proper k-coloring these graphs unless the ETH is false.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Graph coloring
Keywords
  • graph class recognition
  • proper coloring
  • geometric intersection graph
  • segment intersection graph
  • fine-grained complexity
  • Exponential Time Hypothesis

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