On Compaction and Realizability of Almost Convex Octilinear Representations

2025-11-25

AI Summary

This work addresses the octilinear graph drawing problem, comprising two core subproblems: (i) octilinear realizability—deciding whether a given plane embedding admits a valid octilinear drawing satisfying prescribed angular constraints at vertices and a bound on edge bends; and (ii) octilinear compaction—computing a minimum-area layout among all feasible realizations. We establish that realizability is NP-hard even under “almost-convex” vertex representations, and further prove strong inapproximability. When parameterized by the number of reflex angles, we design both an FPT algorithm and an XP algorithm, thereby establishing tight complexity bounds. Our approach integrates parameterized complexity analysis, graph-theoretic modeling, and computational geometry techniques. This work provides the first systematic theoretical framework and practical algorithmic toolkit for geometric graph drawing under octilinear constraints.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Investigating NP-hard realizability of octilinear graph drawings with convexity constraints
• Proving hardness of octilinear compaction without polynomial-time approximation schemes
• Developing FPT and XP algorithms for constrained octilinear drawing problems

Innovation

• Proves Octilinear Realizability NP-hard for non-convex faces
• Strengthens Octilinear Compaction hardness without PTAS
• Provides FPT and XP algorithms for reflex corners