A Supnick matrix or Supnick array – named after Fred Supnick of the City College of New York, who introduced the notion in 1957 – is a Monge array which is also a symmetric matrix.
Contents
Mathematical definition
A Supnick matrix is a square Monge array that is symmetric around the main diagonal.
An n-by-n matrix is a Supnick matrix if, for all i, j, k, l such that if
then
and also
A logically equivalent definition is given by Rudolf & Woeginger who in 1995 proved that
A matrix is a Supnick matrix iff it can be written as the sum of a sum matrix S and a non-negative linear combination of LL-UR block matrices.The sum matrix is defined in terms of a sequence of n real numbers {αi}:
and an LL-UR block matrix consists of two symmetrically placed rectangles in the lower-left and upper right corners for which aij = 1, with all the rest of the matrix elements equal to zero.
Properties
Adding two Supnick matrices together will result in a new Supnick matrix (Deineko and Woeginger 2006).
Multiplying a Supnick matrix by a non-negative real number produces a new Supnick matrix (Deineko and Woeginger 2006).
If the distance matrix in a traveling salesman problem can be written as a Supnick matrix, that particular instance of the problem admits an easy solution (even though the problem is, in general, NP hard).