> var sx{a in A, b in B}, binary;
> /* sx[a,b] is sign(sum{c in C} x[a,b,c]) */
> /* in other word, sx[a,b] is logical_or{c in C} x[a,b,c] */
> 
> s.t. foo{a in A, b in B}: sum{c in C} x[a,b,c] <= card(C) * sx[a,b];
> 
> s.t. rooms{a in A}: sum{b in B} sx[a,b] <= r;

Incorrect. The first constraint must be the following:

s.t. foo{a in A, b in B}:
     0 <= card(C) * sx[a,b] - sum{c in C} x[a,b,c] <= card(C) - 1;





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