[Paper Review] Improved Approximation for Guarding Simple Galleries from the Perimeter

James King, David Kirkpatrick|arXiv (Cornell University)|Jan 24, 2010

Computational Geometry and Mesh Generation1 references4 citations

TL;DR

This paper presents an $O(\log\log\mathrm{OPT})$-approximation algorithm for guarding simple polygons using guards placed on the perimeter or at vertices, improving upon the prior $O(\log\mathrm{OPT})$ bound. It achieves this by designing an $\varepsilon$-net finder for hitting set instances derived from the guarding problem and applying the Brönnimann-Goodrich framework, enabling polynomial-time approximation with provable sub-logarithmic performance for simple polygons.

ABSTRACT

We provide an O(log log OPT)-approximation algorithm for the problem of guarding a simple polygon with guards on the perimeter. We first design a polynomial-time algorithm for building epsilon-nets of size O(1/epsilon log log 1/epsilon) for the instances of Hitting Set associated with our guarding problem. We then apply the technique of Bronnimann and Goodrich to build an approximation algorithm from this epsilon-net finder. Along with a simple polygon P, our algorithm takes as input a finite set of potential guard locations that must include the polygon's vertices. If a finite set of potential guard locations is not specified, e.g. when guards may be placed anywhere on the perimeter, we use a known discretization technique at the cost of making the algorithm's running time potentially linear in the ratio between the longest and shortest distances between vertices. Our algorithm is the first to improve upon O(log OPT)-approximation algorithms that use generic net finders for set systems of finite VC-dimension.

Motivation & Objective

• To develop a polynomial-time approximation algorithm for guarding simple polygons with guards restricted to the perimeter or vertices, improving upon the $O(\log\mathrm{OPT})$ bound.
• To design an $\varepsilon$-net finder with size $O\left(\frac{1}{\varepsilon}\log\log\frac{1}{\varepsilon}\right)$ for the hitting set instances arising from the guarding problem.
• To apply the Brönnimann-Goodrich framework to convert the $\varepsilon$-net finder into a full approximation algorithm with improved approximation ratio.
• To handle cases where guard locations are not pre-specified by using a discretization technique at the cost of running time dependence on the ratio $\Delta$ of longest to shortest vertex distances.

Proposed method

• Construct an $\varepsilon$-net finder for hitting set instances derived from the art gallery problem with perimeter or vertex guards, achieving size $O\left(\frac{1}{\varepsilon}\log\log\frac{1}{\varepsilon}\right)$.
• Apply the Brönnimann-Goodrich framework, which uses $\varepsilon$-nets to derive an $O(\log\log\mathrm{OPT})$-approximation algorithm for the guarding problem.
• Use a known discretization technique when no finite set of guard locations is provided, ensuring the algorithm remains polynomial in $n$ and $\Delta$.
• Analyze the structure of visibility fragments and their types (1–6) to bound the number of fragments per level, proving that only a constant number of fragments per level can be 'fruitful' and thus control the net size.
• Establish that only fragments of types (3)–(6) can be fruitful and that type (6) fragments can have at most two fruitful children, limiting the growth of the visibility structure.
• Leverage the VC-dimension properties of the visibility set system to ensure the $\varepsilon$-net construction is efficient and the resulting algorithm runs in polynomial time.

Experimental results

Research questions

• RQ1 Can the approximation ratio for guarding simple polygons with perimeter or vertex guards be improved beyond $O(\log\mathrm{OPT})$?
• RQ2 Is it possible to achieve a $o(\log\log\mathrm{OPT})$ approximation for perimeter guards, or even a constant-factor approximation matching the APX-hardness result?
• RQ3 What is the true VC-dimension of the set system for perimeter guards, and can the upper bound of 23 be improved?
• RQ4 Can the same $O(\log\log\mathrm{OPT})$ approximation be extended to point guards, despite their more complex visibility structures?

Key findings

• The paper achieves an $O(\log\log\mathrm{OPT})$-approximation for guarding simple polygons using vertex or perimeter guards, representing the first improvement over the $O(\log\mathrm{OPT})$ bound for this setting.
• An $\varepsilon$-net finder is constructed with size $O\left(\frac{1}{\varepsilon}\log\log\frac{1}{\varepsilon}\right)$, which is suboptimal compared to standard $O(\frac{1}{\varepsilon})$ nets but sufficient for the improved approximation.
• The algorithm runs in polynomial time in $n$ and the number of potential guard locations, and remains polynomial in $n$ and $\Delta$ when no finite guard set is given.
• The analysis shows that at most two fragments per level can be fruitful, and the number of fragments at level $i$ that see a given point is at most $4i$, which bounds the VC-dimension-related complexity.
• The paper establishes that the VC-dimension of the set system for perimeter guards is at least 5 and possibly as high as 6, though the upper bound of 23 for point guards applies a fortiori to perimeter guards.
• The work leaves open the possibility of a constant-factor approximation for perimeter guards, matching the APX-hardness result of Eidenbenz, and suggests that improving the VC-dimension upper bound for perimeter guards may be easier than for point guards.