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Crossing Number is NP-hard for Constant Path-width (and Tree-width)

Petr Hliněný, Liana Khazaliya

TL;DR

This work establishes NP-hardness of computing the exact crossing number $cr(G)$ even when the input graph has bounded path-width $12$ (and tree-width $9$). The authors design a SAT-to-Crossing-Number reduction using a gadget framework (Frame, Variable Gadgets, and clause-encoding cells) and a weighted-edge scheme that enforces rigid crossing patterns, effectively simulating a grid while keeping width constant. They prove that a SAT instance is satisfiable if and only if the constructed graph admits a drawing with at most $k$ crossings, with $k$ carefully chosen as a polynomial function of the instance size. The result closes the question of whether bounded-width graph classes admit tractable crossing-number computations (unless P = NP) and motivates exploring other restricted parameters beyond vertex covers and tree-width.

Abstract

Crossing Number is a celebrated problem in graph drawing. It is known to be NP-complete since 1980s, and fairly involved techniques were already required to show its fixed-parameter tractability when parameterized by the vertex cover number. In this paper we prove that computing exactly the crossing number is NP-hard even for graphs of path-width 12 (and as a result, even of tree-width 9). Thus, while tree-width and path-width have been very successful tools in many graph algorithm scenarios, our result shows that general crossing number computations unlikely (under P!=NP) could be successfully tackled using bounded width of graph decompositions, which has been a 'tantalizing open problem' [S. Cabello, Hardness of Approximation for Crossing Number, 2013] till now.

Crossing Number is NP-hard for Constant Path-width (and Tree-width)

TL;DR

This work establishes NP-hardness of computing the exact crossing number even when the input graph has bounded path-width (and tree-width ). The authors design a SAT-to-Crossing-Number reduction using a gadget framework (Frame, Variable Gadgets, and clause-encoding cells) and a weighted-edge scheme that enforces rigid crossing patterns, effectively simulating a grid while keeping width constant. They prove that a SAT instance is satisfiable if and only if the constructed graph admits a drawing with at most crossings, with carefully chosen as a polynomial function of the instance size. The result closes the question of whether bounded-width graph classes admit tractable crossing-number computations (unless P = NP) and motivates exploring other restricted parameters beyond vertex covers and tree-width.

Abstract

Crossing Number is a celebrated problem in graph drawing. It is known to be NP-complete since 1980s, and fairly involved techniques were already required to show its fixed-parameter tractability when parameterized by the vertex cover number. In this paper we prove that computing exactly the crossing number is NP-hard even for graphs of path-width 12 (and as a result, even of tree-width 9). Thus, while tree-width and path-width have been very successful tools in many graph algorithm scenarios, our result shows that general crossing number computations unlikely (under P!=NP) could be successfully tackled using bounded width of graph decompositions, which has been a 'tantalizing open problem' [S. Cabello, Hardness of Approximation for Crossing Number, 2013] till now.

Paper Structure

This paper contains 16 sections, 5 theorems, 1 equation, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Drawings
  4. Tree-width and Path-width
  5. Hardness Reduction
  6. High-level Idea
  7. Auxiliary Graphs
  8. Variable Gadgets.
  9. The Frame.
  10. The Full Reduction
  11. Drawings Claims
  12. Proofs for \ref{['subsec:claims']} (Drawings Claims)
  13. Correctness
  14. On Path-width of the Resulting Instance
  15. Proofs for \ref{['subsec:width']} (On Path-width of the Resulting Instance)
  16. ...and 1 more sections

Key Result

Theorem 1

The problem to decide whether a graph $G$ can be drawn with at most $k$ crossings, for $G$ and $k$ on the input, is -complete even when $G$ is required to be of path-width at most $12$ and of tree-width at most $9$.

Figures (2)

  • Figure 1: Auxiliary Graphs.
  • Figure 3: An example instance of $\SAT$: $\mathcal{V}=\{x_1,x_2,x_3,x_4,x_5\}$ and $\mathcal{C}=\{(x_1 \vee \overline{x_2} \vee x_4 \vee \overline{x_5}),(\overline{x_1} \vee \overline{x_3} \vee x_5),(x_2 \vee x_3 \vee \overline{x_4})\}$, and a reduction to Crossing Number of the depicted graph $G$. The two drawings illustrate assignment two assignments of the variables.

Theorems & Definitions (5)

  • Theorem 1: cf. \ref{['thm:mainred']}
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • Proposition 5