Partial Searchlight Scheduling is Strongly PSPACE-Complete

TL;DR

This paper introduces the Partial Searchlight Scheduling Problem, a generalization of the classic search problem, and proves it is strongly PSPACE-complete, highlighting its computational difficulty even in restricted polygonal environments.

Contribution

It establishes the PSPACE-completeness of the Partial Searchlight Scheduling Problem, extending complexity results to orthogonal polygons and using novel reductions from nondeterministic constraint logic.

Findings

The problem is strongly PSPACE-complete in general environments.

The problem remains PSPACE-complete even for rectangular search regions.

The reduction uses properties of nondeterministic constraint logic machines.

Abstract

The problem of searching a polygonal region for an unpredictably moving intruder by a set of stationary guards, each carrying an orientable laser, is known as the Searchlight Scheduling Problem. Determining the computational complexity of deciding if the polygon can be searched by a given set of guards is a long-standing open problem. Here we propose a generalization called the Partial Searchlight Scheduling Problem, in which only a given subregion of the environment has to be searched, as opposed to the entire area. We prove that the corresponding decision problem is strongly PSPACE-complete, both in general and restricted to orthogonal polygons where the region to be searched is a rectangle. Our technique is to reduce from the "edge-to-edge" problem for nondeterministic constraint logic machines, after showing that the computational power of such machines does not change if we allow "asynchronous" edge reversals (as opposed to "sequential").