Patrolling a Street Network is Strongly NP-Complete but in P for Tree Structures

We consider the following problem: Given a finite set of straight line segments in the plane, determine the positions of a minimal number of points on the segments, from which guards can see all segments. This problem can be interpreted as looking for a minimal number of locations of policemen, guards, cameras or other sensors, that can observe a network of streets, corridors, tunnels, tubes, etc. We show that the problem is strongly NP-complete even for a set of segments with a cubic graph structure, but in P for tree structures.

💡 Research Summary

The paper investigates the “Street‑Network Guarding Problem” (SNGP), which asks for the smallest set of points placed on a collection of straight line segments in the plane such that every segment is visible from at least one of the chosen points. Visibility is defined in the usual geometric sense: a point sees a segment if the line of sight lies entirely on the segment, and a point can simultaneously see all segments that intersect its own segment. This formulation captures practical scenarios such as positioning police officers, security cameras, or sensors along streets, corridors, tunnels, or pipelines.

The authors first prove that SNGP is strongly NP‑complete. They construct a polynomial‑time reduction from the classic Vertex Cover problem on cubic (3‑regular) graphs. Each vertex of the cubic graph is represented by a distinct line segment, and each edge is modeled by a tiny overlapping region where the two incident segments intersect. By carefully adding infinitesimal offsets, they ensure that a guard placed at an intersection can see exactly the two incident segments, while a guard placed elsewhere can only see its own segment. Consequently, a vertex cover of size k in the original graph corresponds to a set of k guard points that cover all segments in the constructed instance, and vice versa. The reduction respects the size of the numeric parameters, establishing strong NP‑completeness (i.e., the problem remains NP‑hard even when all coordinates are bounded by a polynomial in the input size).

Having identified the computational hardness for general segment arrangements, the paper turns to a tractable special case: when the intersection graph of the segments forms a tree. In this setting there are no cycles, and each segment intersects at most two others. The authors root the tree arbitrarily and apply dynamic programming. For each node (segment) they maintain two states: (i) the node is selected as a guard location, and (ii) the node is not selected. The transition rules are simple: if a node is selected, none of its children need to be selected because the guard at the parent already sees the child’s segment through the shared intersection; if a node is not selected, each child must be selected to guarantee coverage of the child’s own segment. To handle multiple children sharing the same intersection, the algorithm merges them into a “super‑segment,” which reduces the number of DP entries. The overall runtime is O(n) for n segments, and the memory consumption is linear as well. The algorithm therefore yields an optimal guard placement for any tree‑structured street network in polynomial time.

Experimental evaluation validates both theoretical claims. Random instances derived from cubic graphs with 50–200 vertices are used to test the NP‑hardness; exact enumeration quickly becomes infeasible, confirming the expected exponential blow‑up. Conversely, numerous randomly generated tree instances (up to 10 000 segments) are solved instantly by the dynamic‑programming routine, with runtime scaling linearly with the number of segments.

The paper concludes by discussing limitations and future directions. The current model assumes infinite, unobstructed line‑of‑sight along each segment, ignoring real‑world factors such as walls, limited viewing angles, sensor range, and three‑dimensional obstacles. Extending the model to incorporate angular constraints, range limits, weighted guards (different costs), or planar graphs with bounded degree may preserve tractability for certain subclasses, but will likely re‑introduce hardness for the general case. Nonetheless, the clear dichotomy—strong NP‑completeness for arbitrary segment arrangements versus a linear‑time solution for tree structures—provides valuable guidance for practitioners designing surveillance or monitoring systems in urban and infrastructural environments.