Supnick matrix

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.

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

$1\leq i<k\leq n$ and $1\leq j<l\leq n$

then

$a_{ij}+a_{kl}\leq a_{il}+a_{kj}\,$

and also

$a_{ij}=a_{ji}.\,$

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 {α_i}:

$S=[s_{ij}]=[\alpha _{i}+\alpha _{j}];\,$

and an LL-UR block matrix consists of two symmetrically placed rectangles in the lower-left and upper right corners for which a_ij = 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).

References

• Supnick, Fred (July 1957). "Extreme Hamiltonian Lines". The Annals of Mathematics (2nd series). 66 (1): 179–201.
• Woeginger, Gerhard J. (June 2003). "Computational Problems without Computation" (PDF). Nieuwarchief. 5 (4): 140–147.
• Vladimir G. Deineko and Gerhard J. Woeginger (2006): 'Some problems around travelling salesmen, dart boards, and euro-coins', appeared in the Bulletin of the European Association for Theoretical Computer Science (EATCS), Number 90, October 2006, ISSN 0252-9742, pages 43 - 52. See online edition (PDF).