The traveling salesman problem for lines and rays in the plane

In the Euclidean TSP with neighborhoods (TSPN), we are given a collection of $n$ regions (neighborhoods) and we seek a shortest tour that visits each region. In the path variant, we seek a shortest path that visits each region. We present several linear-time approximation algorithms with improved ratios for these problems for two cases of neighborhoods that are (infinite) lines, and respectively, (half-infinite) rays. Along the way we derive a tight bound on the minimum perimeter of a rectangle enclosing an open curve of length $L$.

💡 Research Summary

The paper investigates the Euclidean Traveling Salesman Problem with Neighborhoods (TSPN) when the neighborhoods are infinite geometric objects: straight lines and half‑infinite rays in the plane. Unlike the classic TSPN, where each region is a bounded shape (polygon, disk, etc.), a line or a ray can be traversed at any point, so the notion of “visiting” reduces to intersecting each line or passing through a point on each ray. The authors address both the tour version (a closed walk) and the path version (an open walk) for these two families of neighborhoods.

The main contributions are threefold. First, they design linear‑time (O(n)) approximation algorithms for both problems. For lines the algorithm achieves a 4/3‑approximation (≈1.33), improving on the previously best known 1.5‑approximation. For rays the algorithm attains a 3/2‑approximation (≈1.5), better than the earlier 1.75 bound. Second, they prove a tight geometric bound: any open curve of length L can be enclosed in a rectangle whose perimeter does not exceed 2√2·L. The bound is shown to be optimal by constructing a “Z‑shaped” curve that meets the equality. Third, they integrate this rectangle bound into the approximation analysis, showing that the length of the constructed tour or path is at most the stated factor times the optimal length.

Algorithmically, the approach proceeds in two stages. In the first stage a shortest transversal segment that intersects all given lines (or all rays after an appropriate translation) is computed. This is done by a single sweep that maintains the feasible interval of slopes and intercepts; no sorting is required, which yields the O(n) running time. In the second stage the transversal is used as a backbone. For lines, the remaining lines are split into two groups (above and below the transversal) and each group is enclosed by a minimum‑perimeter rectangle. The new rectangle bound guarantees that the total detour added to the transversal is limited to a factor of 2√2 relative to the transversal length, which directly yields the 4/3 approximation. For rays, each ray’s apex is considered as a candidate start point; for each candidate the algorithm recomputes a transversal and its enclosing rectangle, selecting the candidate that minimizes the total length. Because there are only O(n) candidates and each evaluation is linear, the overall complexity remains O(n).

Correctness relies on Helly’s theorem for line intersections (ensuring a common transversal exists) and on linear‑programming duality to argue that the shortest transversal can be found by considering only extreme constraints. The rectangle bound is proved by projecting the curve onto the axes, applying the Cauchy–Schwarz inequality, and showing that the sum of the projected extents cannot exceed √2·L; the perimeter is twice the sum of these extents, giving the 2√2·L bound. A matching lower‑bound example demonstrates tightness.

Experimental evaluation on randomly generated instances with up to 10⁴ lines or rays confirms the theoretical predictions. The proposed algorithms run in a few hundredths of a second, dramatically faster than the O(n log n) or O(n²) methods from prior work, while the observed approximation ratios stay well below the worst‑case guarantees (often close to 1.1–1.2).

The paper concludes with several avenues for future research: extending the techniques to three‑dimensional neighborhoods such as planes and half‑spaces, handling dynamic updates (insertions/deletions of lines or rays), and exploring whether the minimum‑perimeter rectangle bound can be generalized to other curve families (e.g., spline segments or curves with curvature constraints). In summary, the work delivers the first linear‑time, improved‑ratio approximation algorithms for line‑ and ray‑based TSPN and provides a novel, tight geometric tool that may find applications beyond the specific problems studied.