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On Compaction and Realizability of Almost Convex Octilinear Representations

Henry Förster, Giacomo Ortali, Lena Schlip

TL;DR

The paper investigates the computational complexity of octilinear drawing problems under constrained representations. It strengthens NP-hardness and inapproximability results for Octilinear Realizability and Octilinear Compaction, respectively, under limited non-convexity, while delivering positive algorithmic results: Octilinear Realizability is fixed-parameter tractable in the number of reflex corners, and Octilinear Compaction admits an XP algorithm parameterized by the number of diagonals, when the outer face has few reflex corners. Central techniques include shadow graphs and flow-based convex formulations, plus plumbing of planarity-preserving extensions to achieve parameterized algorithms. The findings clarify the boundary between tractable and intractable instances, guiding practical drawing tools under limited non-convexity and diagonal complexity. The work advances the theoretical understanding of octilinear representations and provides actionable parameterized approaches for real-world diagram layouts.

Abstract

Octilinear graph drawings are a standard paradigm extending the orthogonal graph drawing style by two additional slopes (+1 and -1). We are interested in two constrained drawing problems where the input specifies a so-called representation, that is: a planar embedding; the angles occurring between adjacent edges; the bends along each edge. In Orthogonal Realizability one is asked to compute any orthogonal drawing satisfying the constraints, while in Orthogonal Compaction the goal is to find such a drawing using minimum area. While Orthogonal Realizability can be solved in linear time, Orthogonal Compaction is NP-hard even if the graph is a cycle. In contrast, already Octilinear Realizability is known to be NP-hard. In this paper we investigate Octilinear Realizability and Octilinear Compaction problems. We prove that Octilinear Realizability remains NP-hard if at most one face is not convex or if each interior face has at most 8 reflex corners. We also strengthen the hardness proof of Octilinear Compaction, showing that Octilinear Compaction does not admit a PTAS even if the representation has no reflex corner except at most 4 incident to the external face. On the positive side, we prove that Octilinear Realizability is FPT in the number of reflex corners and for Octilinear Compaction we describe an XP algorithm on the number of edges represented with a +1 or -1 slope segment (i.e., the diagonals), again for the case where the representation has no reflex corner except at most 4 incident to the external face.

On Compaction and Realizability of Almost Convex Octilinear Representations

TL;DR

The paper investigates the computational complexity of octilinear drawing problems under constrained representations. It strengthens NP-hardness and inapproximability results for Octilinear Realizability and Octilinear Compaction, respectively, under limited non-convexity, while delivering positive algorithmic results: Octilinear Realizability is fixed-parameter tractable in the number of reflex corners, and Octilinear Compaction admits an XP algorithm parameterized by the number of diagonals, when the outer face has few reflex corners. Central techniques include shadow graphs and flow-based convex formulations, plus plumbing of planarity-preserving extensions to achieve parameterized algorithms. The findings clarify the boundary between tractable and intractable instances, guiding practical drawing tools under limited non-convexity and diagonal complexity. The work advances the theoretical understanding of octilinear representations and provides actionable parameterized approaches for real-world diagram layouts.

Abstract

Octilinear graph drawings are a standard paradigm extending the orthogonal graph drawing style by two additional slopes (+1 and -1). We are interested in two constrained drawing problems where the input specifies a so-called representation, that is: a planar embedding; the angles occurring between adjacent edges; the bends along each edge. In Orthogonal Realizability one is asked to compute any orthogonal drawing satisfying the constraints, while in Orthogonal Compaction the goal is to find such a drawing using minimum area. While Orthogonal Realizability can be solved in linear time, Orthogonal Compaction is NP-hard even if the graph is a cycle. In contrast, already Octilinear Realizability is known to be NP-hard. In this paper we investigate Octilinear Realizability and Octilinear Compaction problems. We prove that Octilinear Realizability remains NP-hard if at most one face is not convex or if each interior face has at most 8 reflex corners. We also strengthen the hardness proof of Octilinear Compaction, showing that Octilinear Compaction does not admit a PTAS even if the representation has no reflex corner except at most 4 incident to the external face. On the positive side, we prove that Octilinear Realizability is FPT in the number of reflex corners and for Octilinear Compaction we describe an XP algorithm on the number of edges represented with a +1 or -1 slope segment (i.e., the diagonals), again for the case where the representation has no reflex corner except at most 4 incident to the external face.

Paper Structure

This paper contains 17 sections, 15 theorems, 1 equation, 13 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hardness Results
  4. Review of the NP-hardness Construction of Bekos et al. DBLP:journals/algorithmica/BekosFK19
  5. Approximability of Octilinear Compaction
  6. Para-NP-Hardness of Octilinear Realizability
  7. Parametrized Results
  8. Flow algorithm for the convex case.
  9. An FPT Algorithm for Octilinear Realizability
  10. Shadow Graph and Representation
  11. Planar Extensions and FPT Algorithm
  12. An XP Algorithm for Octilinear Compaction
  13. Open Problems
  14. Omitted Details from Section \ref{['sec:nphardnessliterature']}
  15. Omitted Details from Section \ref{['sec:inapprox']}
  16. ...and 2 more sections

Key Result

theorem thmcountertheorem

There is no $\frac{9}{4}$-approximation for Octilinear Compaction even if $\omega = 4$ unless P=NP.

Figures (13)

  • Figure 1: Gadgets used in the reduction from $3$-SAT to Octilinear Realizability DBLP:journals/algorithmica/BekosFK19 for $\ell_u=1$: (a) Propagation, (b) rerouting, (c) copy, (d)--(e) variable and (f) clause gadget. (d) and (e) show $x=\top$ and $x=\bot$, respectivel.y
  • Figure 2: Example of the reduction descibed in DBLP:journals/algorithmica/BekosFK19 with formula $(x_1\lor x_2 \lor x_3) \land (\neg x_1 \lor \neg x_2 \lor x_3) \land (\neg x_1 \lor x_2 \lor x_3)$ when $\ell_u= 1$ and the satisfying assignment $x_1=\bot$, $x_2 = x_3 = \top$ with our modifications (red and blue) as described in the proof of Theorem \ref{['thm:approximation']}. A clean version without our adjustments is in Appendix \ref{['app:hard']}.
  • Figure 3: A representation $\mathcal{R}$ and auxiliary flow networks $G^\rightarrow$ (blue) and $G^\downarrow$ (red). The value of flow along all thicker arcs is 2, the value of flow along all other arcs is 1. The positions on the grid corresponds to the solution yielded.
  • Figure 4: The construction of $G'$ and $\mathcal{R'}$.
  • Figure 5: (a) Graph $G$ and its representation $\mathcal{R}$. (b) Shadow graph $G'$ and shadow representation $\mathcal{R}'$ obtained from Graph $G$ and its representation $\mathcal{R}$. (c) Planar extension $\mathcal{R}"$ of $\mathcal{R}'$.
  • ...and 8 more figures

Theorems & Definitions (26)

  • theorem thmcountertheorem
  • proof : Sketch
  • corollary thmcountercorollary
  • theorem thmcountertheorem
  • proof : Sketch
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • ...and 16 more