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01 integer programming
Posted:
Jun 16, 1996 7:28 AM


[In the question below, the poster would like to know the thinnest covering of the finite set {0,1}^k by Hamming balls of radius 1. In particular, he would like a practical algorithm to find such a covering for k <= 16.  Greg]
I'm interesting to solve this 01 integer programming problem, that is a minimal covering set problem:
Min sum(Xj), j:=1,...,n An X >= 1, where An is a matrix recursive like that:
 A1 = 1
 A(n1 I(n1) An = I(n1) A(n1)
where In is de identity matrix of order n.  n= 2**k, where k:= 1, 2, 3, 4......  Xj = (0,1).  An(i,j) = (0,1)
Example: n=8=2**3
Min X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8
1 1 1 0 1 0 0 0 X1 1 1 1 0 1 0 1 0 0 X2 1 1 0 1 1 0 0 1 0 X3 1 0 1 1 1 0 0 0 1 X4 1 1 0 0 0 1 1 1 0 X5 >= 1 0 1 0 0 1 1 0 1 X6 1 0 0 1 0 1 0 1 1 X7 1 0 0 0 1 0 1 1 1 X8 1
That is: X1 + X2 + X3 + X5 >= 1 X1 + X2 + X4 + X6 >= 1 X1 + X3 + X4 + X7 >= 1 + X2 + X3 + X4 + X8 >= 1 X1 + X5 + X6 + X7 >= 1 + X2 + X5 + X6 + X8 >= 1 + X3 + X5 + X7 + X8 >= 1 + X4 + X6 + X7 + X8 >= 1
The matrix An is symetric, and his determin sometimes is 0.
I'm practising and solving this problem with the aditiv algorithm of Balas implemented by me in Turbo Pascal lenguage, and his results about the complexity of time dont satisfy me. This problem for n=32=2**5 was executed in 6 hours aproximately in one Pentium 100 Mhz. I'm interesting to solve the maximum number of variables about this problem (from n=64 to 16384).
All suggestions or recomendations about this problem, or bibliography to see will be gratefully.
Excuse me for my poor English.
Regards. I hope your mail or news.
Miguel Angel Camara Facultad de Cc. Matemticas Universidad Complutense de Madrid Spain
EMail: mcamaral@madrid.idec.es



