Originally posted by: Bryanr

Hello,

I'm wondering how to model the negation in cplex Java-api. I need to define XNOR constraints over binary variables in my model. Any advice ? Thanks a lot
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Originally posted by: T_O

XNOR over 2 binaries is just equivalence, so you can omit one of the variables (replacing it by the other one everywhere) or add a constraint like x1 - x2 = 0.

Concerning more than 2 variables, we first of all have to ensure that we use the same definition of XNOR. In the following, I will assume that it is check for even parity (the number of ones is even).

Let x1, ..., xn be the binaries be XNORed. Add a new integer variable z. Then add the following constraint:
x1 + ... + xn - 2z = 0

Best regards,
Thomas
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Originally posted by: Bryanr

Thanks for your help Thomas! As you said for two variables it is straightforward but concerning the multiple variables I was looking for a way to formulate without adding new integer variables to deal with it.
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Originally posted by: SystemAdmin

In order to avoid the auxiliary integer variable, you would need to add all the facets of the XNOR polytope.
For three variables, this is:

x1 - x2 - x3 <= 0
-x1 + x2 - x3 <= 0
-x1 - x2 + x3 <= 0
 x1 + x2 + x3 <= 2

See, for example, my thesis at http://opus4.kobv.de/opus4-zib/frontdoor/index/index/docId/1112, Section 14.9, page 233.

Unfortunately, the number of facets of the polytope increases exponentially with the number of variables, see Section 14.12, pages 237-238.

Tobias
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