Rectangle Tiling Binary Arrays

TL;DR

The paper addresses rectangle tiling of binary $n\times n$ arrays by partitioning into at most $p$ tiles to minimize the maximum tile weight, and also studies the dual DRTile problem. It introduces a boundary/shadow framework and uses linear programming duality to derive tight approximation guarantees, achieving a linear-time $\left(\tfrac{3}{2}+\tfrac{p^2}{w(A)}\right)$-approximation for RTile, with a $\tfrac{3}{2}$-approximation in the regime $\tfrac{p^2}{w(A)}$ negligible and a controlled extension to the generalized case. The results are shown to be tight with respect to the existing lower bound $L=\lceil w(A)/p \rceil$, and the methods extend to the dual problem, to multidimensional variants, and to a reduction-based approach for DRTile, yielding practical, linear-time tiling strategies for applications such as load balancing and data partitioning. Overall, the work significantly improves the best-known approximation ratios for binary rectangle tiling and provides a coherent, globally symmetric LP-based framework for both the primal and dual problems across dimensions.

Abstract

The problem of rectangle tiling binary arrays is defined as follows. Given an $n \times n$ array $A$ of zeros and ones and a natural number $p$, our task is to partition $A$ into at most $p$ rectangular tiles, so that the maximal weight of a tile is minimized. A tile is any rectangular subarray of $A$. The weight of a tile is the sum of elements that fall within it. We present a linear $(O(n^2))$ time $(\frac{3}{2}+\frac{p^2}{w(A)})$-approximation algorithm (where $\frac{p^2}{w(A)} < \frac{1}{2}$) for this problem, where $w(A)$ denotes the weight of the whole array $A$. This improves on the previously known approximation with the ratio $2$. The result is best possible in the following sense. The algorithm employs the lower bound of $L=\lceil \frac{w(A)}{p} \rceil$, which is the only known and used bound on the optimum in all algorithms for rectangle tiling. We prove that a better approximation factor for the binary \RTILE cannot be achieved using $L$, because there exist arrays, whose every partition contains a tile with weight at least $(\frac{3}{2}+\frac{p^2}{w(A)})L$. We also consider the dual problem of rectangle tiling for binary arrays, where we are given an upper bound on the weight of the tiles, and we have to cover the array $A$ with the minimum number of non-overlapping tiles. Both problems have natural extensions to $d$-dimensional versions, for which we provide analogous results.

Figures (5)

• Figure 1: An array with column $A[4]$ as the only boundary
• Figure 2: An array with one horizontal boundary containing the subarrays $x_2$ and $x_1$ and one vertical boundary containing the subarrays $x_3$, $x_1$ and $x_4$
• Figure 3: An array with one vertical and one horizontal boundary
• Figure 4: An array with two vertical and two horizontal boundaries
• Figure 5: The empty squares denote a value of $0$, while the ones are colored black. $(a)$ On tiling this array with $3$ tiles, one tile will always contain $5$ ones, giving an approximation factor of $\frac{5}{3}$. $(b)$$4$-crosses placed in an array for proving an approximation lower bound of $\frac{3}{2}$.

Theorems & Definitions (21)

• Definition 1
• Definition 2
• Example 3
• Example 4
• Example 5
• Lemma 5
• Lemma 5
• Lemma 5
• Example 7
• Example 8