We're given a $N\times M$ grid, and we want to cover all coordinates in the greedy by rectangles of size $\le k$.
Consider the following greedy algorithm. At each iteration, it chooses a rectangle of area $\le k$ which covers as many uncovered elements as possible. If more than one such rectangle exists, it chooses one arbitrarily.
Can we prove a bound on the approximation ratio of this algorithm?
For example, if $M=1,N=6,k=3$ (i.e., we wish to cover a length $6$ line with intervals of size $3$), the optimal (minimizing the number of rectangles in the cover) picks a rectangle covering $[1,2,3]$ and another one covering $[4,5,6]$.
The greedy algorithm, on the other hand, may first choose the interval $3,4,5$, and then proceed with $1,2,3$ and finally $4,5,6$.
The number of rectangles used by the algorithm is $3$, while the optimum is $2$, thus the approximation ratio of the example is $1.5$.
I have a proof that the approximation ratio of the 1D case ($M=1$) is $2$ (and it is tight).
What is the approximation ratio of the general $N\times M$ grid algorithm?
I'm well aware that it is probably possible to solve the problem optimally in linear time with a different algorithm, but in my application the rectangles are generated arbitrarily (e..g, think of a stores who wish to serve as many customers as possible in a grid) and are not ours to choose.