Realizing Planar Linkages in Polygonal Domains

Abstract: A linkage $\mathcal{L}$ consists of a graph $G=(V,E)$ and an edge-length function $\ell$. Deciding whether $\mathcal{L}$ can be realized as a planar straight-line embedding in $\mathbb{R}<sup>2$ with edge length $\ell(e)$ for all $e \in E$ is $\exists\mathbb{R}$-complete [Abel et al., JoCG'25], even if $\ell \equiv 1$, but a considerable part of $\mathcal{L}$ is rigid. In this paper, we study the computational complexity of the realization question for structurally simpler, less rigid linkages inside an open polygonal domain $P$, where the placement of some vertices may be specified in the input. We show XP-membership and W[1]-hardness with respect to the size of $G$, even if $\ell \equiv 1$ and no vertex positions are prescribed. Furthermore, we consider the case where $G$ is a path with prescribed start and end position and $\ell \equiv 1$. Despite the absence of any rigid components, we obtain NP-hardness in general, and provide a linear-time algorithm for arbitrary $\ell$ if $G$ has only three edges and $P$ is convex.

• The paper establishes that PLR is in XP by formulating the problem as an existential real algebra system with fixed parameters for vertex coordinates.
• It proves W[1]-hardness and NP-hardness for unit-length and linear linkages, respectively, through reductions from GRID TILING and Planar Monotone 3-SAT.
• A linear-time algorithm for three-edge linkages in convex domains demonstrates how domain geometry can simplify otherwise intractable configurations.

Realizing Planar Linkages in Polygonal Domains: Complexity and Algorithms

Problem Formulation and Context

The paper addresses the Planar Linkage Realizability (PLR) problem. Given a graph $G = (V, E)$ and a positive edge-length function $\ell: E \rightarrow \mathbb{R}^+$, a configuration is a straight-line drawing of $G$ in $\mathbb{R}^2$ where each edge has its prescribed length and no edges cross. In the setting examined here, the realization must further be fully contained within a specified open polygonal domain $P \subset \mathbb{R}^2$. The primary computational question is whether such a planar realization exists, possibly with fixed positions for a subset of vertices.

PLR generalizes classic linkage realizability within the plane (known to be $\exists\mathbb{R}$-complete even for unit-length edges [ADD+.Wnc.2025]) by incorporating environmental constraints via the domain $P$. The primary motivation for this study is to understand how obstacles or domain geometry impact the tractability of geometric constraint systems, especially for combinatorially simpler but less rigid linkages, such as paths (linear linkages).

Main Complexity Results

XP-Membership with Respect to Graph Size

The first significant contribution is that PLR is in XP with respect to $|V|$, the size of the linkage graph, even when edge lengths are unrestricted. Specifically, the decision problem is expressible as an existential theory of the reals ($\exists\mathbb{R}$) formula with a bounded number of variables (twice the number of vertices, encoding $x$ and $\ell: E \rightarrow \mathbb{R}^+$0 coordinates). Satisfiability of such polynomial systems can be decided in time $\ell: E \rightarrow \mathbb{R}^+$1 for $\ell: E \rightarrow \mathbb{R}^+$2 variables [GJ.SSP.1988]. The explicit formula construction enforces (1) prescribed vertex positions, (2) edge length constraints, (3) planarity (non-crossing edges) using determinants, and (4) containment of the drawing within $\ell: E \rightarrow \mathbb{R}^+$3. This establishes fixed-parameter tractability with parameter $\ell: E \rightarrow \mathbb{R}^+$4, but the running time is non-parameterized polynomial.

$\ell: E \rightarrow \mathbb{R}^+$5-Hardness Parameterized by Graph Size

The above upper bound is tight in a parameterized sense: the paper proves that PLR is W[1]-hard with respect to $\ell: E \rightarrow \mathbb{R}^+$6, even for unit-length linkages and polygonal domains with polynomial complexity. This is achieved via a parameterized reduction from GRID TILING, a canonical W[1]-hard problem [Cygan2015]. The reduction realizes the encoding of discrete choices into the position of linkage vertices constrained to small regions (grid squares with encoded tiles), leveraging geometric gadgets to simulate the grid tiling compatibility constraints. The construction demonstrates that no combinatorial parameterization (e.g., vertex count, even number of rigid components) leads to an FPT algorithm unless $\ell: E \rightarrow \mathbb{R}^+$7, settling the parameterized complexity of PLR for bounded-size linkages.

$\ell: E \rightarrow \mathbb{R}^+$8-Hardness for Linear Unit-Length Linkages

A central technical contribution is the demonstration that even for linear linkages (i.e., $\ell: E \rightarrow \mathbb{R}^+$9 is a simple path) with unit-length edges, and with only the start and end positions constrained, PLR remains NP-hard. The reduction is from Planar Monotone 3-SAT [DBK.OBS.2012]. The construction carefully orchestrates a polygonal domain $G$0 (possibly with holes) and a sequence of geometric gadgets representing variables, clauses, and edge connections. These gadgets enforce a one-to-one correspondence between realizable linkage configurations and truth assignments, ensuring that satisfying assignments to the logical formula correspond to collision-free, planar, edge-length respecting embeddings of the path from the prescribed start to end points.

Linear-Time Algorithm in Convex Domains for Three-Edge Linkages

Contrasting this hardness, the paper provides an $G$1 linear-time algorithm for the case where $G$2 is a path of length three (four vertices) and $G$3 is a convex polygon with $G$4 vertices. The key geometric insight is that such linkages have just one degree of freedom and the planarity and containment constraints can be checked efficiently by enumerating possible boundary contacts and testing feasibility, reducing verification to a finite set of geometric configurations. This result holds for arbitrary positive edge lengths.

Theoretical and Practical Implications

Parameterized and Classical Intractability

The establishment of both W[1]-hardness (for unit-length graphs) and NP-hardness (even for linear/unit-linkages with only terminal constraints) highlights the computational intractability of geometric constraint realization in the presence of obstacles. Importantly, the reduction shows that rigidity per se is not responsible for hardness; even completely flexible paths inherit computational complexity through their interaction with the embedding domain.

As a consequence, algorithmic approaches for linkage realization in practical CAD/CAM, robotics, or molecular applications must leverage features of the domain, such as low complexity of $G$5, rather than structural simplicity of the linkage alone.

The Role of Domain Geometry

The complexity and algorithmic boundary for PLR is shown to critically depend on the structure of $G$6. For convex domains and bounded chain length, tractable algorithms exist. However, the presence of holes or concavities is crucial for encoding hard combinatorial problems, as evidenced by the NP-hardness construction. This has direct consequences for geometric motion planning and rigidity theory.

The results motivate further investigation into parameterizations by domain complexity (number of holes, reflex vertices, etc.), with the conjecture that fixed-parameter tractability might be recovered for linkages in domains with bounded topological or combinatorial complexity.

Algorithmic Techniques

The explicit handling and encoding of realizability as systems of polynomial equalities and inequalities provides a template for direct application of symbolic computation and real algebraic methods—even as the worst-case complexity remains high. The geometric gadget constructions for encoding logical relations have broader applicability to reductions for other geometric realization-type problems.

Directions for Further Research

• Algorithmic meta-theorems for PLR in domains with bounded holes or treewidth.
• Hardness for simple polygons: the present hardness leverages holes; proving NP-hardness for realization in simple (hole-free) polygons remains open.
• Extension to other constraints: e.g., linkages with repeated lengths, dichotomous edges [Angelini2025], or in higher-dimensions.
• Efficient algorithms for small linkages: Extending the tractability for length-three linkages to longer chains, possibly via configuration space analysis or motion planning techniques.

Conclusion

This work precisely delineates the boundary between tractability and intractability for planar linkage realization in constrained (polygonal) domains. While fixed-parameter and polynomial-time algorithms exist for chains in convex environments, general cases—even for structurally simple paths—are intractable due to the expressive power of geometric constraint encoding via the domain. The findings have significant ramifications for geometric constraint solving, motion planning, and the algorithmic theory of rigidity, and guide avenues for further research on parameterized algorithms and domain-sensitive complexity classification.

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References:

• "Who needs crossings?: Noncrossing linkages are universal, and deciding (global) rigidity is hard" [ADD+.Wnc.2025]
• "Parameterized Algorithms" [Cygan2015]
• "Solving Systems of Polynomial Inequalities in Subexponential Time" [GJ.SSP.1988]
• "Optimal Binary Space Partitions for Segments in the Plane" [DBK.OBS.2012]
• "Geometric Realizations of Dichotomous Ordinal Graphs" [Angelini2025]