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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 ≤ i < k ≤ n {\displaystyle 1\leq i<k\leq n} and 1 ≤ j < l ≤ n {\displaystyle 1\leq j<l\leq n}

then

a i j + a k l ≤ a i l + a k j {\displaystyle a_{ij}+a_{kl}\leq a_{il}+a_{kj}\,}

and also

a i j = a j i . {\displaystyle 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 {αi}:

S = [ s i j ] = [ α i + α j ] ; {\displaystyle 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 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).

• Supnick, Fred (July 1957). "Extreme Hamiltonian Lines". Annals of Mathematics. Second Series. 66 (1): 179–201. doi:10.2307/1970124. JSTOR 1970124.
• Woeginger, Gerhard J. (June 2003). "Computational Problems without Computation" (PDF). Nieuwarchief. 5 (4): 140–147.
• Deineko, Vladimir G.; Woeginger, Gerhard J. (October 2006). "Some problems around travelling salesmen, dart boards, and euro-coins" (PDF). Bulletin of the European Association for Theoretical Computer Science. 90. EATCS: 43–52. ISSN 0252-9742.