An Efficient Algorithm for Computing High-Quality Paths amid Polygonal Obstacles

We study a path-planning problem amid a set $\mathcal{O}$ of obstacles in $\mathbb{R}^2$, in which we wish to compute a short path between two points while also maintaining a high clearance from $\mathcal{O}$; the clearance of a point is its distance from a nearest obstacle in $\mathcal{O}$. Specifically, the problem asks for a path minimizing the reciprocal of the clearance integrated over the length of the path. We present the first polynomial-time approximation scheme for this problem. Let $n$ be the total number of obstacle vertices and let $\varepsilon \in (0,1]$. Our algorithm computes in time $O(\frac{n^2}{\varepsilon ^2} \log \frac{n}{\varepsilon})$ a path of total cost at most $(1+\varepsilon)$ times the cost of the optimal path.

💡 Research Summary

The paper addresses the problem of planning a high‑quality path for a point robot moving in the plane among a set 𝒪 of polygonal obstacles. The quality measure combines path length with clearance: for a path γ, the cost is defined as µ(γ)=∫₀¹ 1/ clr(γ(τ)) dτ, where clr(p) is the Euclidean distance from p to the nearest obstacle. This cost penalizes paths that come too close to obstacles while still rewarding short routes, thus capturing a trade‑off that many practical applications (e.g., surgical robotics, autonomous driving) require.

Previous work either computes exact shortest collision‑free paths (O(n log n) algorithms) or enforces a hard clearance bound δ, but none provides a polynomial‑time approximation for the combined metric. An earlier approximation scheme by Wein et al. depends on a parameter Λ that can be exponential in the input size, making it impractical.

The authors present the first polynomial‑time approximation scheme (PTAS) for this metric. Given n obstacle vertices and an error parameter ε∈(0,1], their algorithm runs in O(n² ε⁻² log (n/ε)) time and returns a path whose cost is at most (1+ε) times the optimal cost π(s,t).

The algorithm proceeds in three main stages:

1. Voronoi Diagram Construction and Refinement
   The Voronoi diagram 𝓥 of the obstacle features (vertices and edges) partitions the free space into cells V(o). Each cell is star‑shaped and has O(1) complexity. The authors refine each cell by adding line segments from every feature to its Voronoi vertices and by inserting a segment from the feature to the point on the cell boundary that minimizes clearance. The resulting refined diagram ˜𝓥 still has O(n) complexity, but each refined cell T has a very regular structure: one external edge κ_T (a parabola or line segment) and two internal edges α_T and β_T (line segments). This regularity is crucial for the subsequent analysis.

2. Well‑Behaved Paths and Anchor Points
   Inside a refined cell, the authors prove that for any two boundary points p and q, there exists a “well‑behaved” path γ(p,q) whose cost is at most a constant factor (≤11) of the optimal sub‑path cost π(p,q). The construction uses an “anchor point” w* on the internal edge β_T that maximizes clearance while minimizing the cost of the sub‑path λ(p;w)=p→w→η_w, where η_w is the maximal constant‑clearance arc inside the cell. Lemma 3.2 shows that µ(γ(p,q))≤11 π(p,q). Moreover, the anchor point can be found in O(1) time because the cost function is monotone and has a closed form.

3. Graph Construction with an ε‑Spanner
   Using the well‑behaved path property, the algorithm samples O(1) points on each edge of ˜𝓥 and adds the anchor points, yielding an O(n)‑size “coarse” graph G₁. A simple shortest‑path computation on G₁ gives a 2‑approximation. To achieve a (1+ε)‑approximation, the authors further sample O(ε⁻¹ log ε⁻¹) points uniformly along each edge and, within each cell T, build a sparse ε‑spanner that connects these samples while preserving distances up to a (1+ε) factor. The spanner construction leverages the fact that any optimal sub‑path inside a cell stays on a constant‑clearance arc, allowing the use of standard geometric spanner techniques. The final graph G has O(n² ε⁻² log (n/ε)) vertices and edges.

The overall time complexity is dominated by the spanner construction and the subsequent Dijkstra run, both bounded by O(n² ε⁻² log (n/ε)). The authors also provide a detailed geometric analysis proving that the optimal path consists of circular arcs, logarithmic spirals, straight segments, and pieces of Voronoi edges, each starting and ending on the refined diagram’s boundaries. This structural insight justifies the limited sampling and guarantees the approximation factor.

In summary, the paper delivers a theoretically sound and practically efficient PTAS for a realistic path‑quality metric that balances length and safety. It advances the state of the art by removing the exponential dependence on the cost of the Voronoi diagram, offering a solution that scales polynomially with the number of obstacle vertices and the desired precision. The techniques—refined Voronoi cells, well‑behaved sub‑paths, and ε‑spanners—are likely to be applicable to other multi‑criteria motion‑planning problems, and they open avenues for extensions to three‑dimensional environments, dynamic obstacles, and real‑time implementations.