The $k$-Colorable Unit Disk Cover Problem

The 3-CDUDC problem, to the best of our knowledge, was first studied by Biedl et al., [BBL19]. They gave a 2-approximation algorithm in $O(nm^{25})$ time for the 3-CDUDC problem. Their approach first partitions the plane into horizontal strips, solves the problem for every strip optimally, then returns the union of solutions of all strips. To solve the problem for any strip they show that at most a constant number of disks of an optimal solution intersect any vertical line. Based on this, they define a directed acyclic graph such that there exists a path from source to a destination corresponding to this optimal solution. In this paper, we attempt to improve upon this impractical $O(nm^{25})$ running time. Our approach, however, focuses on the specific geometric properties that arise from the dual conditionals of the problem statement. Although both of the approaches, initially, begin by dividing the plane, we recognize a unique bound that exists in our need to bound the colorability and provide a novel solution in the same regard.

A notion of conflict-free coloring (CF-coloring) was introduced by Even et al., [ELRS03]. and Smorodinsky [S03]. In the CF-coloring problem we are given a set of points (representing client locations) and a set of base stations, the objective is to assign colors (representing frequencies) to the base stations such that any client lying within the range of at least one base station is covered by the base station whose color is different from the colors of the other base stations covering the client, and the number of colors used should be as minimum as possible. Here, the range of base stations is modeled as regions e.g., disks or other geometric objects. Even et al., [ELRS03] proved that $O(\log n)$ colors are always sufficient to CF-color a set of disks in the plane, and in the worst case, $\Omega(\log n)$ colors are required. Note that this CF-coloring of disks is different from our notion of $k$-colorable disk cover of points. In the former overlapping disks may be given the same color if they dont share a client, whereas in the $k$-CDUDC overlapping disks must be colored with distinct colors regardless of whether they cover a common client. A generalization of CF-coloring is called a $k$-fault-tolerant CF-coloring. Cheilaris et al., [CGRS14] presented a polynomial-time $(5-\frac{2}{k})$-approximation algorithm for the $k$-fault-tolerant CF-coloring in 1-dimensional space. Horev et al., [HKS10] proved that $O(k\log n)$ colors are sufficient for any set of $n$ disks in the plane. For dynamic CF-coloring and results on CF-coloring of other geometric objects, we refer to [BM19] and references therein.

A related problem of the $k$-CDUDC problem in the literature is the Discrete Unit Disk Cover (DUDC) problem. In the DUDC problem, we are given a set $P$ of $n$ points and a set $D$ of $m$ unit disks, our goal is to select as the smallest number of disks from $D$ as possible such that the union of these selected disks covers all points in $P$. As in the $k$-CDUDC, here also, the sets $P$ and $D$ can be considered as representing a set of wireless clients and a set of base stations or towers, respectively. The DUDC problem is NP-hard and is a very well studied one. There is a polynomial time approximation scheme (PTAS) with impractical running time for this problem [MR10]. The current best approximation algorithm with reasonable running time is $(9+\epsilon)$ for any $\epsilon>0$ [BAD15]. However, a series of approximation algorithms have been proposed for this problem by various authors over the past two decades, and a complete survey on this can be found in [FL12]. When a line segment is used to represent a potential wireless client, the DUDC problem becomes a Line Segment Disk Cover (LSDC) problem. In a similar line, there is another variant of the DUDC problem, a Rectangular Region Cover (RRC) problem, in which all the continuous set of points lying in a rectangular region represent wireless clients.  All the available results for the DUDC problem also extend to the LSDC and RRC problems [B18], with slightly different running time. We also extend our results for the $k$-CDUDC problem to solve the colorable variants of the LSDC and RRC problems, namely, the $k$-CLSDC and $k$-CRRC problems.

Here, our algorithm is based on partitioning the plane containing points into a grid and then determining bound on the number of unit disks that can participate in any $k$-colorable covering of points lying within any square of the grid. We first define a grid of width $\tau$ units that partitions the plane into squared regions. Each of these squared regions is a grid cell with a size $\tau\times\tau$. For simplicity assume no point of $P$ lies on the boundary of these grid cells. Let us associate a unique ID $id_{\cal C}$ to each grid cell ${\cal C}$ as follows; let $p=(x_{p},y_{p})$ be a point in ${\cal C}$ and $\tau$ be the grid width, then $id_{\cal C}=(\lfloor\frac{x_{p}}{\tau}\rfloor,\lfloor\frac{y_{p}}{\tau}\rfloor)$, (see Fig. 1). Note that each grid cell has a unique ID associated with it but multiple points can be associated with the same ID (if they lie within the corresponding grid cell). Let $id_{{\cal C}_{1}}$ and $id_{{\cal C}_{2}}$ be any two arbitrary grid cells with base points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ respectively. We define the greater than operator for an ID as follows: $(id_{{\cal C}_{1}}>id_{{\cal C}_{2}})\iff((x_{1}>x_{2})\land(y_{1}>y_{2}))\lor((x_{1}=x_{2})\land(y_{1}>y_{2}))\lor((x_{1}>x_{2})\land(y_{1}=y_{2}))$. Note that our defintion of $id_{\cal C}$ implies that the grid cells are indexed from bottom-left to top-right, this operator simply indicates the order of iteration that is followed by our algorithm. We move left to right row-wise starting from the bottom-left corner cell. A pre-defined order is essential to our handing-over logic at line 11 of Algorithm 1.

Given a grid cell ${\cal C}^{\prime}$, the following lemma provides a bound on the cardinality of the $k$-colorable unit disks covering all points lying within the grid cell ${\cal C}^{\prime}$. Let $D_{{\cal C}^{\prime}}\subseteq D$ be the set of $k$-colorable unit disks covering all points of $P$ lying within the grid cell ${\cal C}^{\prime}$. The proof of the lemma is based on the observation that determining this bound is the same as determining a maximum number of disjoint unit disks that could potentially intersect ${\cal C}^{\prime}$.

The outline of our algorithm (Algorithm 1) for computing a cover $D^{\prime}\subseteq D$ of the points $P$ is as follows. We first partition the rectangular region containing the objects in $D$ and $P$ into individual grid cells of size $\tau\times\tau$. By utilizing the bound obtained in Observation 1 (for e.g., for $\tau=2$ the actual bound is in Lemma 1) we compute a $k$-colorable cover of the points lying in each grid cell, in an exhaustive manner. To ensure that there is no conflict in the overall aggregate solution, we use a handing-over logic. Only disks of any particular grid cell cover centered within the same grid cell are colored with the associated color set. If a disk is required to be a part of this grid cell cover, but is centered in another grid cell, it is handed-over to that grid cell. Based on the grid width $\tau$ and the diameter of the disk, we then define a coloring scheme $\chi$ that assigns a color to each disk in the union $D^{\prime}$ of all the individual grid cell covers computed. Finally, we return the pair $(D^{\prime},\chi)$. Since the diameter of the disks is fixed to be two units, the approximation factor of the algorithm is implied by the choice of the value $\tau$. If the value of $\tau$ is 2, then a unit disk can participate in the $k$-colorable covers of points lying in four adjacent grid cells. Hence, we prove that Algorithm 1 is a 4-approximate algorithm (see Theorem LABEL:theorem-2.2). Later, we show that by varying the grid width $\tau$, which results in a unit disk participating in more than four individual grid cell covers, we can obtain a family of algorithms with approximation factors corresponding to the choice of the value of $\tau$ (see Subsection LABEL:generalize2.2).

We now define any coloring function that assigns colors to disks to be conflict-free if for any pair of non-disjoint disks (i.e., overlapping disks) the colors assigned to them are different.