On exact covering with unit disks

Abstract

We study the problem of covering a given point set in the plane by unit disks so that each point is covered exactly once. We prove that 17 points can always be exactly covered. On the other hand, we construct a set of 657 points where an exact cover is not possible.

Figures (13)

• Figure 1: Left: Primal solution (exact covering set). Right: Dual solution (exact hitting set).
• Figure 2: Proof idea of the Extension Argument \ref{['lem: Kozma method']}. We extend the red disjoint disk covering of the non-boundary points by adding a new orange disk at each uncovered boundary point. The new disks may overlap each other or existing disks.
• Figure 3: How to find the fourth generalized boundary point in the proof of Theorem \ref{['lem: large triangles have four generalized boundary points']}. The bulldozers are $\mathop{\mathrm{BD}}\nolimits\left(X, -0.75\right)$ (medium orange) and $\mathop{\mathrm{BD}}\nolimits\left(X, t_{\max} + \varepsilon\right)$ (light orange), with $t_{\max} = 0.5$. The generalized boundary point $\mathbf{b} = -0.25\space1.5\space \in X'$ is in dark purple and the disk $D\left(\varepsilon\right) = -0.250.5 + \varepsilon + B^{2}$ in dark orange. The grey arrows represent the intuition of finding $\mathbf{b}$ by moving the bulldozer "upwards" until it reaches a point of $X'$. The point $\mathbf{b}'$ is on the boundary of $\mathop{\mathrm{BD}}\nolimits\left(X, t_{\max}\right)$, so it is also a generalized boundary point of $X$ which we could have chosen instead of $\mathbf{b}$.
• Figure 4: An example that illustrates the need for the complicated definition of the bulldozer. The grey disks are an exact cover of $X \left\backslash\, \left\{\mathbf{b}\right\}\right.$ while the orange disk $D\left(\varepsilon\right)$ (see the proof of Lemma \ref{['lem: large triangles have four generalized boundary points']}) is from the bulldozer. Left: The point $\mathbf{b} = \mathbf{y}$ has smallest $y$-coordinate among all points in $X'$. In this case, $\mathbf{b}$ cannot be covered by a disk without $\mathbf{x}$ or $\mathbf{v}^{2}$ being covered by two disks. (The faint brown disk at the top right shows that $\mathbf{b}$ cannot be covered in this way by approaching from another edge of $T$.) Right: The definition of the bulldozer fixes the problem in Figure \ref{['fig: bulldozer 3']}. The bulldozer finds a different point $\mathbf{b} = \mathbf{x}$, and this choice of $\mathbf{b}$ results in a different disjoint cover $\mathscr{D}'$ of $X \left\backslash\, \left\{\mathbf{b}\right\} \right.\!$. If $\mathbf{b}$ is not covered by a disk in $\mathscr{D}'$ then $\mathscr{D}' \cup \left\{D\left(\varepsilon\right)\right\}$ is an exact cover of $X$. If $\mathbf{b}$ is already covered by a disk in $\mathscr{D}'$, as in Figure \ref{['fig: bulldozer 4']}, then $\mathscr{D}'$ is an exact cover of $X$.
• Figure 5: A comparison of $D_{i, j}$ and $\mathop{\mathrm{BD}}\nolimits_{i, j}$. The unit disks $D_{i, j}$ intersect at the orthocenter $\mathbf{h}$ and $T \subseteq D_{1, 2} \cup D_{1, 3} \cup D_{2, 3}$. Left: If $R_{T} > 1$ (in this case $\frac{8}{7} \approx 1.071$), then the bulldozers are smaller than the unit disks and hence cover $T'$ (Lemma \ref{['lem: medium triangles have four generalized boundary points']}), but a single unit disk (light blue) cannot cover $T'$. Right: If $R_{T} < 1$ (in this case $\frac{45}{56} \approx 0.804$), then the bulldozers are larger than the unit disks and cannot cover $T'$, but one unit disk (light blue) covers $T'$.

Theorems & Definitions (58)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Lemma 2.2: Extension Argument
• proof
• Remark 2.3
• Definition 2.4
• Lemma 2.5: Generalized Extension Argument
• proof : Proof of the Generalized Extension Argument, Part 1