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I wonder about the following decision problem :

Instance: We consider a $n\times p$ matrix $M$ of zeros and ones, and two integers $N$ and $k$.

Question: is it possible to cover all the ones of the matrix with $N$ squares of side length $k$?

It is possible that the squares cover some of the zeros in the matrix, and they can overlap with each other.

I get the impression that it is $\textsf{NP}$-complete (it is clearly $\textsf{NP}$), I find some similarities with Grid covering by rectangles but can't find a good reduction.

Might I get some insight on the problem?

Thanks.

EDIT: After days of reflexion, these are the ideas I've got:

I try to create a reduction from a Satisfiability problem with the following approach.

Given $\varphi = \bigwedge\limits_{i=1}^m C_i$ where $\forall i \in [\![1,m]\!]$, $C_i = (\ell_{i1}\vee \ell_{i2}\vee\ell_{i3})$ and $\ell_{ij}$ is either $x$ or $\neg x$ where $x\in\mathcal{V} = \{x_1,…,x_n\}$. I create the matrix $A$ with dimensions $(2m+2)\times (6n-1)$ defined by:

  • $\forall j \in [\![1,n]\!], a_{1,6(j-1)+3} = a_{2m+2,6(j-1)+3} = 1$;
  • $\forall i,j\in [\![1,m]\!]\times [\![1,n]\!]$, for $k = 2i - 1$ and $k = 2i$ and for $p = 6(j-1) +2$, $p = 6(j-1) + 3$ and $p = 6(j-1) + 4$, $a_{kp} = 1$;
  • all other coefficients are set to 0.

Here's a visual representation of the matrix:

matrix construction

With this matrix, if, for a variable $j$, we allow the use of $2m-1$ squares of size $2\times 2$, we can either cover all of (except extremities) the column $6(j-1)+1$ or the column $6(j-1) + 5$, but not both and not a mix between them. This can simulate a boolean variable!

The idea I got then was to put ones in the column $6(j-1)+1$ save if $x_j$ appears in the clause $C_i$ in which case, $a_{2i-1, 6(j-1)+1}$ and $a_{2i,6(j-1)+1}$ stay to zero (and do the same thing similarly in the column $6(j-1)+5$ with the $\neg x$ litteral).

If we then allow the use of $2n-1$ squares for each clause $C_i$, we need one of the variables to be set correctly in the clause to cover all ones in lines $2i-1$ and $2i$. The problem with this reduction is that you can "gain" some squares in a clause with multiples variables set correctly to change the disposition of a variable elsewhere.

My second idea was to create a reduction from $\texttt{ONE-IN-THREE-SAT}$, in which we ask if there is an assignation to $\varphi$ which satisfies exactly one litteral per clause, or even its restriction where we can suppose that $\varphi$ contains no negations, but I couldn't find something satisfying, so I'd like some help again!

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