Question

I have to partition a multiset into two sets who sums are equal. For example, given the multiset:

1 3 5 1 3 -1 2 0

I would output the two sets:

1) 1 3 3
2) 5 -1 2 1 0

both of which sum to 7.

I need to do this using Z3 (smt2 input format) and “Linear Arithmetic Logic”, which is defined as:

formula : formula /\ formula | (formula) | atom
   atom : sum op sum
     op : = | <= | <
    sum : term | sum + term
   term : identifier | constant | constant identifier

I honestly don’t know where to begin with this and any advice at all would be appreciated.

Regards.

Answer 1

Here is an idea:

1- Create a 0-1 integer variable c_i for each element. The idea is c_i is zero if element is in the first set, and 1 if it is in the second set. You can accomplish that by saying that 0 <= c_i and c_i <= 1.

2- The sum of the elements in the first set can be written as 1*(1 - c_1) + 3*(1 - c_2) + ... +

3- The sum of the elements in the second set can be written as 1*c1 + 3*c2 + ...