Rectangle Tiling Binary Arrays
Pratik Ghosal, Syed Mohammad Meesum, Katarzyna Paluch

TL;DR
This paper introduces a linear-time approximation algorithm for rectangle tiling binary arrays that improves the approximation ratio from 2 to approximately 1.5, providing near-optimal solutions and extending results to dual and higher-dimensional problems.
Contribution
The paper presents a new linear-time approximation algorithm with a better ratio for rectangle tiling binary arrays, and proves the optimality of this ratio using a specific lower bound.
Findings
Achieves a $(rac{3}{2}+rac{p^2}{w(A)})$-approximation ratio in $O(n^2)$ time.
Improves upon the previous approximation ratio of 2 for the rectangle tiling problem.
Extends the results to dual and $d$-dimensional versions of the problem.
Abstract
The problem of rectangle tiling binary arrays is defined as follows. Given an array of zeros and ones and a natural number , our task is to partition into at most rectangular tiles, so that the maximal weight of a tile is minimized. A tile is any rectangular subarray of . The weight of a tile is the sum of elements that fall within it. We present a linear time -approximation algorithm (where ) for this problem, where denotes the weight of the whole array . This improves on the previously known approximation with the ratio . The result is best possible in the following sense. The algorithm employs the lower bound of , which is the only known and used bound on the optimum in all algorithms for rectangle tiling. We prove that a better…
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