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Sometimes Two Irrational Guards are Needed

Lucas Meijer, Tillmann Miltzow

TL;DR

This work proves that an optimal guarding of a simple monotone polygon with two guards may necessitate irrational coordinates, extending prior results that required three irrational guards and that one guard can be placed rationally. The authors construct a rational-coordinate polygon with two guard segments arranged to force a unique pair of irrational guard positions, and they prove that no two rational guards can guard the polygon. The approach leverages a detailed geometric construction with pockets and guard segments, together with an examination of the corresponding existential theory of the reals (ER/ETR) encoding to justify the necessity of irrational coordinates. The result sharpens our understanding of when irrational guard placements are unavoidable and provides a concrete benchmark for testing algorithms and discretization schemes in the art gallery problem.

Abstract

In the art gallery problem, we are given a closed polygon $P$, with rational coordinates and an integer $k$. We are asked whether it is possible to find a set (of guards) $G$ of size $k$ such that any point $p\in P$ is seen by a point in $G$. We say two points $p$, $q$ see each other if the line segment $pq$ is contained inside $P$. It was shown by Abrahamsen, Adamaszek, and Miltzow that there is a polygon that can be guarded with three guards, but requires four guards if the guards are required to have rational coordinates. In other words, an optimal solution of size three might need to be irrational. We show that an optimal solution of size two might need to be irrational. Note that it is well-known that any polygon that can be guarded with one guard has an optimal guard placement with rational coordinates. Hence, our work closes the gap on when irrational guards are possible to occur.

Sometimes Two Irrational Guards are Needed

TL;DR

This work proves that an optimal guarding of a simple monotone polygon with two guards may necessitate irrational coordinates, extending prior results that required three irrational guards and that one guard can be placed rationally. The authors construct a rational-coordinate polygon with two guard segments arranged to force a unique pair of irrational guard positions, and they prove that no two rational guards can guard the polygon. The approach leverages a detailed geometric construction with pockets and guard segments, together with an examination of the corresponding existential theory of the reals (ER/ETR) encoding to justify the necessity of irrational coordinates. The result sharpens our understanding of when irrational guard placements are unavoidable and provides a concrete benchmark for testing algorithms and discretization schemes in the art gallery problem.

Abstract

In the art gallery problem, we are given a closed polygon , with rational coordinates and an integer . We are asked whether it is possible to find a set (of guards) of size such that any point is seen by a point in . We say two points , see each other if the line segment is contained inside . It was shown by Abrahamsen, Adamaszek, and Miltzow that there is a polygon that can be guarded with three guards, but requires four guards if the guards are required to have rational coordinates. In other words, an optimal solution of size three might need to be irrational. We show that an optimal solution of size two might need to be irrational. Note that it is well-known that any polygon that can be guarded with one guard has an optimal guard placement with rational coordinates. Hence, our work closes the gap on when irrational guards are possible to occur.
Paper Structure (33 sections, 1 theorem, 8 equations, 13 figures, 1 table)

This paper contains 33 sections, 1 theorem, 8 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Organization.
  3. Discussion
  4. Minimum number of irrational guards.
  5. Grid Approximation.
  6. Intuitive Understanding.
  7. Test Cases.
  8. Technical Depth.
  9. Irrational Boundary Guards.
  10. Integer Coordinates.
  11. Special Polygons.
  12. Art Gallery Problem
  13. Exact Algorithmic Results.
  14. Practical Difficulty.
  15. Explaining the Discrepancy.
  16. ...and 18 more sections

Key Result

Theorem 1

There exists a simple monotone polygon with rational coordinates, such that there is only one way of guarding this polygon optimally with two guards. Those two guards have irrational coordinates.

Figures (13)

  • Figure 1: Lucas and Till guarding the polygon with just the two of them...
  • Figure 2: Any triangulation of a simple polygon can be three-colored. At least one of the color classes has at most $\lfloor n/3\rfloor$ vertices. This color class also guards the entire polygon, as every triangle is incident to all three colors Fisk78a.
  • Figure 3: We may restrict the guards to lie on a dense grid. This may make the optimal solution worse.
  • Figure 4: Left: The dark green region is added to the visibility polygon. Right: The orange region is removed from the visibility polygon.
  • Figure 5: Our final polygon: it has a core (gray), three quadrilateral pockets (blue), and four narrow triangular pockets (yellow).
  • ...and 8 more figures

Theorems & Definitions (1)

  • Theorem 1