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

TL;DR

This paper investigates the complexity of placing minimal surveillance points on street networks, proving NP-completeness for general cases but polynomial-time solvability for tree-structured networks.

Contribution

It establishes the NP-completeness of the guard placement problem for general street networks and provides a polynomial-time solution for tree-structured networks.

Findings

NP-complete for general street networks with cubic graph structure

Polynomial-time algorithm for tree-structured networks

Highlights complexity differences based on network topology

Abstract

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.