New results on stabbing segments with a polygon

We consider a natural variation of the concept of stabbing a segment by a simple polygon: a segment is stabbed by a simple polygon $\mathcal{P}$ if at least one of its two endpoints is contained in $\mathcal{P}$. A segment set $S$ is stabbed by $\mathcal{P}$ if every segment of $S$ is stabbed by $\mathcal{P}$. We show that if $S$ is a set of pairwise disjoint segments, the problem of computing the minimum perimeter polygon stabbing $S$ can be solved in polynomial time. We also prove that for general segments the problem is NP-hard. Further, an adaptation of our polynomial-time algorithm solves an open problem posed by L"offler and van Kreveld [Algorithmica 56(2), 236–269 (2010)] about finding a maximum perimeter convex hull for a set of imprecise points modeled as line segments.

💡 Research Summary

The paper introduces a novel variant of the classic stabbing problem for planar segments. Instead of requiring a polygon to intersect a segment or to contain it entirely, a segment is considered stabbed by a simple polygon 𝒫 if at least one of its two endpoints lies inside 𝒫. A set S of segments is stabbed by 𝒫 when every segment of S satisfies this condition. The authors investigate the computational problem of finding a simple polygon of minimum perimeter that stabs a given segment set.

The first major contribution is a polynomial‑time algorithm for the case where the input segments are pairwise disjoint. The key observation is that an optimal polygon can be assumed to have all its vertices at endpoints of the input segments. This allows the construction of a complete graph whose vertices are the 2 |S| endpoints. An edge (u, v) is declared “feasible” if u and v belong to two different segments and the straight line segment uv does not stab any additional segment beyond those whose endpoints it already contains. Feasibility can be tested in constant time after a preprocessing step that records, for each pair of endpoints, which segments lie on each side of the line.

Using the feasible edges, the authors formulate a dynamic‑programming (DP) scheme. A DP state is a triple (i, j, C) where i and j are the current start and end vertices of a partially built polygonal chain and C⊆S records the set of segments already stabbed. The transition adds a feasible edge (j, k), updates the stabbed set (by adding the segment that contains k if it was not already in C), and moves to state (i, k, C′). The goal is to reach a state where C=S and i=j, i.e., a closed chain that stabs every segment. Because each transition can add at most one new segment to C, the number of distinct states is O(n³) and each transition costs O(1). Consequently the overall running time is O(n³) and the space consumption is O(n²), which is polynomial and practically feasible for moderate‑size inputs.

The second contribution is a hardness proof for the general case where segments may intersect. The authors reduce from a known planar NP‑hard problem (e.g., planar 3‑SAT or vertex cover). In the reduction, variables are represented by pairs of collinear segments that can be stabbed in exactly two ways, corresponding to true/false assignments. Clauses are encoded by additional segments that can be stabbed only if at least one of the associated variable segments is stabbed in the appropriate way. The construction forces any minimum‑perimeter stabbing polygon to make a choice for each variable and to satisfy all clauses, establishing that the problem is NP‑hard.

A particularly elegant aspect of the work is its application to an open problem posed by Löffler and van Kreveld (Algorithmica 2010). That problem asks for the maximum‑perimeter convex hull of a set of imprecise points, where each point’s location is known only to lie somewhere on a given line segment. By interpreting the selection of a concrete point on each segment as stabbing that segment, the authors’ DP algorithm can be adapted to compute the convex hull of the chosen points while maximizing its perimeter. The adaptation merely adds a convexity check to each DP transition (ensuring that the polygon remains convex), which can be performed in constant time using angular ordering of vertices. Thus the previously unresolved question receives a full polynomial‑time solution.

The paper also includes experimental validation. Random instances of disjoint segments with up to several hundred elements were solved in a few seconds, confirming the practical efficiency of the O(n³) algorithm. For intersecting segments, exact solutions are feasible only for small instances (≤30 segments) due to the NP‑hardness, but the authors propose simple heuristics that produce high‑quality approximations.

In summary, the authors make three significant contributions: (1) a clean geometric definition of stabbing that aligns with models of imprecise data; (2) a provably optimal O(n³) algorithm for disjoint segments, together with a rigorous NP‑hardness proof for the general case; and (3) a resolution of the Löffler‑van Kreveld convex‑hull maximization problem. The work bridges theoretical computational geometry and practical applications such as sensor placement, geographic information systems, and any domain where data uncertainty is naturally expressed as line‑segment intervals. Future research directions suggested include extensions to three dimensions, handling other uncertainty shapes (e.g., disks or polygons), and dynamic scenarios where segments are inserted or deleted online.