Simpler and Faster Contiguous Art Gallery

TL;DR

This paper introduces a significantly simpler and faster polynomial-time algorithm for the contiguous art gallery problem, improving upon previous solutions in efficiency and clarity.

Contribution

It presents a concise, self-contained algorithm with improved runtime complexity for solving the contiguous art gallery problem.

Findings

New algorithm runs in O(k n^2 log^2 n) time

Algorithm is simpler and more efficient than previous solutions

Provides a practical approach for guard placement in polygons

Abstract

The contiguous art gallery problem was introduced at SoCG'25 in a merged paper that combined three simultaneous results, each achieving a polynomial-time algorithm for the problem. This problem is a variant of the classical art gallery problem, first introduced by Klee in 1973. In the contiguous art gallery problem, we are given a polygon P and asked to determine the minimum number of guards needed, where each guard is assigned a contiguous portion of the boundary of P that it can see, such that all assigned portions together cover the boundary of P. The classical art gallery problem is NP-hard and ER-complete, and the three independent works investigated whether this variant admits a polynomial-time solution. Each of these works indeed presented such a solution, with the fastest running in O(k n^5 log n) time, where n denotes the number of vertices of P and k is the size of a minimum guard set covering the boundary of P. We present a solution that is both considerably simpler and significantly faster, yielding a concise and almost entirely self-contained O(k n^2 log^2 n)-time algorithm.