The arctic rank of a Boolean matrix

Among different rank functions on tropical matrices, there is one known as tropical rank which is a lower bound for any other. Here we introduce a new concept (for being opposed to tropical rank, it is called arctic) which gives an upper bound for other ranks. Our definition is based on the perimeter notion previously studied for rank-one matrices. We study the arithmetic behavior of the arctic rank and compare it with that of other rank functions. Finally, we provide an algorithm computing the arctic rank. We show that no algorithm is likely to find the arctic rank in polynomial time by proving that computing the perimeter of a matrix is an NP-hard problem.