# 4/3 Rectangle Tiling lower bound

## TL;DR

This paper proves that approximating the minimal maximum weight rectangle tiling within a factor of 1.33 is NP-hard, improving upon the previous hardness factor of 1.25 for the problem of covering an array with rectangles.

## Contribution

It establishes a tighter NP-hardness bound for the rectangle tiling problem, showing it is harder to approximate than previously known.

## Key findings

- Proves NP-hardness of approximation within factor 1.33
- Improves previous hardness factor of 1.25
- Highlights computational difficulty of optimal rectangle tiling

## Abstract

The problem that we consider is the following: given an $n \times n$ array $A$ of positive numbers, find a tiling using at most $p$ rectangles (which means that each array element must be covered by some rectangle and no two rectangles must overlap) that minimizes the maximum weight of any rectangle (the weight of a rectangle is the sum of elements which are covered by it). We prove that it is NP-hard to approximate this problem to within a factor of \textbf{1$\frac{1}{3}$} (the previous best result was $1\frac{1}{4}$).