On a Problem Posed by Maurice Nivat

Abstract

Consider a $m \times n$ matrix $A$, whose elements are arbitrary integers. Consider, for each square window of size $2 \times 2$, the sum of the corresponding elements of $A$. These sums form a $(m - 1) \times (n-1)$ matrix $S$. Can we efficiently (in polynomial time) restore the original matrix $A$ given $S$? This problem was originally posed by Maurice Nivat for the case when the elements of matrix $A$ are zeros and ones. We prove that this problem is solvable in polynomial time. Moreover, the problem still can be efficiently solved if the elements of $A$ are integers from given intervals. On the other hand, for $2 \times 3$ windows the similar problem turns out to be NP-complete.