Solving a simple TSP using Z3

Question

I am new to SMT solvers. I would like to know that how could I encode a simple TSP problem having 4/6 nodes? I am confused how to set my constraints using the Z3pay APA. Any kind of hint or help would be highly appreciated.

Answer 1

You can phrase your TSP problem as an ILP problem. The question is now how to encode the TSP as an ILP. There are two well known answers: Miller–Tucker–Zemlin and Dantzig–Fulkerson–Johnson.

The basic idea is as follows: Say we have n cities. Let us denote by d_ij the distance between cities i and j and let us denote by x_{ij} the boolean value (0 or 1) whether the TSP contains the edge from i to j. Then finding the smallest tour means

minimize sum_{i,j} x_{ij} d_{ij}

such that the x_{ij} describe a cycle. With those two conditions we get one or more cycles:

sum_{j} x_{ij} = 1 for all i              exactly one outgoing edge per city
sum_{i} x_{ij} = 1 for all j              exactly one ingoing edge per city

Now we have to exclude the case that the solutions comprises multiple cycles. We can add this exponential number of Dantzig–Fulkerson–Johnson conditions:

sum_{i in S} sum_{j in S} x_{ij} < |S| for all proper subsets S of {1, ..., n}

Note that if our solution contains two cycles then for S being the vertex set of one of the cycles then the x_{ij}-sum will be |S|. On the other hand, if there is only one cycle then the x_{ij}-sum will never reach |S|, e.g., if you remove one vertex from {1, ..., n} then the number of edges remaining is n-2, but |S| = n-1.

Of course, an exponential number of constraints is not what we want, so we look for a more clever way to exclude the subcycle cases. And here is where Miller–Tucker–Zemlin jumps in.

A different approach would be to simply ignore the subcycle problem, compute a solution and check whether the solution comprises subcycles. If it does, exclude the solution by adding it as a lazy constraint and repeat until you get a single-cycle solution. The keyword here is lazy constraint.

Answer 2

There is a nice sample that may be useful for you: http://z3.codeplex.com/SourceControl/changeset/view/1235b3ea24d9#examples/python/hamiltonian/hamiltonian.py