[Paper Review] Fast Approximation Algorithms for Art Gallery Problems in Simple Polygons

Dae-Sung Jang, Sunil Kwon|arXiv (Cornell University)|Jan 7, 2011

Computational Geometry and Mesh Generation14 references3 citations

TL;DR

This paper presents O(n³) time approximation algorithms for the vertex guard (VG) and edge guard (EG) problems in simple polygons, improving upon previous O(n⁴) algorithms. By leveraging the fact that only O(n²) visibility region sinks need to be covered—instead of O(n³) visibility regions—the method reduces computation time while maintaining an O(log n) approximation ratio via greedy set cover on visibility sinks.

ABSTRACT

We present approximation algorithms with O(n^3) processing time for the minimum vertex and edge guard problems in simple polygons. It is improved from previous O(n^4) time algorithms of Ghosh. For simple polygon, there are O(n^3) visibility regions, thus any approximation algorithm for the set covering problem with approximation ratio of log(n) can be used for the approximation of n vertex and edge guard problems with O(n^3) visibility sequence. We prove that the visibility of all points in simple polygons is guaranteed by covering O(n^2) sinks from vertices and edges : It comes to O(n^3) time bound.

Motivation & Objective

• To develop faster approximation algorithms for the vertex guard and edge guard problems in simple polygons.
• To reduce the time complexity from O(n⁴) to O(n³) by exploiting structural properties of visibility regions.
• To prove that only O(n²) visibility region sinks need to be covered, rather than all O(n³) visibility regions.
• To maintain an O(log n) approximation ratio while significantly improving runtime efficiency.

Proposed method

• Identify visibility regions in a simple polygon using visibility subdivisions from vertex visibility polygons.
• Construct a dual graph of visibility regions, where edges represent visibility set changes across shared boundaries.
• Define and extract 'sinks'—visibility regions whose visibility set is a subset of all neighboring regions—using the dual graph structure.
• Show that edge guard visibility can be reduced to covering the same set of VG-sinks, due to identical window structures between vertex and edge visibility.
• Compute partially visible edges from each sink using linear-time visibility polygon algorithms.
• Apply a greedy set cover heuristic on the weak visibility sets of edges to find the minimum edge guard set, achieving O(log n) approximation.

Experimental results

Research questions

• RQ1 Can the time complexity of approximation algorithms for the vertex guard problem in simple polygons be reduced below O(n⁴)?
• RQ2 Are there structural properties of visibility regions that allow for a smaller dominating set than the full set of visibility regions?
• RQ3 Can edge guard problems be reduced to vertex guard-style visibility sink covering with equivalent approximation guarantees?
• RQ4 Does the number of visibility region sinks grow more slowly than the total number of visibility regions in simple polygons?
• RQ5 Can the O(n⁴) bottleneck in Ghosh's algorithm be eliminated by focusing only on sinks rather than all visibility regions?

Key findings

• The number of visibility region sinks in a simple polygon is O(n²), significantly fewer than the O(n³) total visibility regions.
• The vertex guard and edge guard problems can be solved by covering only the O(n²) sinks, reducing the problem size.
• The proposed algorithm runs in O(n³) time, improving upon the previous O(n⁴) time complexity for the same approximation ratio.
• The algorithm maintains an O(log n) approximation ratio by applying greedy set cover to the visibility sinks.
• The edge guard problem is reducible to covering the same set of sinks used in the vertex guard problem, due to identical window structures.
• The method is extendable to polygons with holes, but time complexity remains O(n⁵) due to O(n⁴) sinks in such cases.