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
- [math]\displaystyle{ 1\le i \lt k\le n }[/math] and [math]\displaystyle{ 1\le j \lt l\le n }[/math]
then
- [math]\displaystyle{ a_{ij} + a_{kl} \le a_{il} + a_{kj}\, }[/math]
and also
- [math]\displaystyle{ a_{ij} = a_{ji}. \, }[/math]
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}:
- [math]\displaystyle{ S = [s_{ij}] = [\alpha_i + \alpha_j]; \, }[/math]
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).
References
- Supnick, Fred (July 1957). "Extreme Hamiltonian Lines". Annals of Mathematics. Second Series 66 (1): 179–201. doi:10.2307/1970124.
- Woeginger, Gerhard J. (June 2003). "Computational Problems without Computation". Nieuwarchief 5 (4): 140–147. http://www.nieuwarchief.nl/serie5/deel04/jun2003/pdf/woeginger.pdf.
- 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 (EATCS) 90: 43–52. ISSN 0252-9742. http://alexandria.tue.nl/openaccess/Metis211810.pdf.
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