4/3 rectangle tiling lower bound

Abstract

The problem that we consider is the following: given an $n\times n$ array A of positive numbers and a natural number p, 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 1$\frac{1}{3}$ (the previous best result was $1\frac{1}{4}$).