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An Overview of Minimum Convex Cover and Maximum Hidden Set

Reilly Browne

Abstract

We give a review of results on the minimum convex cover and maximum hidden set problems. In addition, we give some new results. First we show that it is NP-hard to determine whether a polygon has the same convex cover number as its hidden set number. We then give some important examples in which these quantities don't always coincide. Finally, We present some consequences of insights from Browne, Kasthurirangan, Mitchell and Polishchuk [FOCS, 2023] on other classes of simple polygons.

An Overview of Minimum Convex Cover and Maximum Hidden Set

Abstract

We give a review of results on the minimum convex cover and maximum hidden set problems. In addition, we give some new results. First we show that it is NP-hard to determine whether a polygon has the same convex cover number as its hidden set number. We then give some important examples in which these quantities don't always coincide. Finally, We present some consequences of insights from Browne, Kasthurirangan, Mitchell and Polishchuk [FOCS, 2023] on other classes of simple polygons.
Paper Structure (18 sections, 16 theorems, 5 equations, 18 figures, 1 table)

This paper contains 18 sections, 16 theorems, 5 equations, 18 figures, 1 table.

Table of Contents

  1. Preliminaries
  2. Hardness Results
  3. Determining the Gap
  4. General position
  5. Exact and Approximation Algorithms
  6. Polygons with Holes
  7. Simple Polygons
  8. Polygons Weakly Visible from a Convex Edge
  9. Using Edges as a Poset
  10. A hidden set without an antichain
  11. Convex cover and hidden set in convex fans and monotone mountains
  12. Rocking chair polygons
  13. Convex cover and hidden set in star-shaped polygons
  14. Convex cover and hidden set in weak visibility polygons
  15. Convex cover and hidden set in simple orthogonal polygons
  16. ...and 3 more sections

Key Result

Theorem 2.1

Deciding if a simple polygon is a homestead polygon or not is NP-hard.

Figures (18)

  • Figure 1: The subgraphs in each of the shermer construction components. Top is a literal unit, left is a consistency checker, right is a clause unit and bottom is the overall structure, indicating we are including the corners.
  • Figure 2: Orthogonal and x-monotone polygon, $M$ which is not a homestead polygon.
  • Figure 3: Showing the remaining regions of $M$ that haven't been ruled out by a starting convex cover.
  • Figure 4: Showing the size 2 convex covers of the remaining region after taking out the strong visibility regions of the purple regions.
  • Figure 5: Showing an induced subgraph of $PVG(M)$ with $cc(G) = 4$ from chromatic number of compliment.
  • ...and 13 more figures

Theorems & Definitions (35)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • ...and 25 more