The Analytic Arc Cover Problem and its Applications to Contiguous Art Gallery, Polygon Separation, and Shape Carving

TL;DR

This paper introduces the analytic arc cover problem and demonstrates polynomial-time solutions for three geometric problems: a contiguous art gallery, polygon separation, and 3D shape carving, advancing computational geometry.

Contribution

It establishes that three geometric problems are in P by analyzing the analytic arc cover problem, a novel interval set cover problem over the circle with infinitely many arcs.

Findings

Contiguous art gallery problem is in P.

Polygon separation for line segments is in P.

Minimizing half-plane cuts for 3D polytopes is in P.

Abstract

We show the following problems are in $\textsf{P}$: 1. The contiguous art gallery problem -- a variation of the art gallery problem where each guard can protect a contiguous interval along the boundary of a simple polygon. This was posed at the open problem session at CCCG '24 by Thomas C. Shermer. 2. The polygon separation problem for line segments -- For two sets of line segments $S_1$ and $S_2$, find a minimum-vertex convex polygon $P$ that completely contains $S_1$ and does not contain or cross any segment of $S_2$. 3. Minimizing the number of half-plane cuts to carve a 3D polytope. To accomplish this, we study the analytic arc cover problem -- an interval set cover problem over the unit circle with infinitely many implicitly-defined arcs, given by a function.