Algorithm for finding subsets subject to union condition

Question

I'm a mathematician and the following came up in my research:

Fix some positive integers $a_0,...,a_n$ and $N$.

Consider subsets $A_i \subset \{0,...,N\}$ where $|A_i|=a_i$, subject to some fixed conditions of the form: $A_{i_0}=A_{j_0}\sqcup A_{j_1}\sqcup ... \sqcup A_{j_t}$. (See an example below).

I want to code an algorithm that takes the values $a_1,...,a_n$, $N$ and the conditions $(i_0, \{j_0,...,j_t\})$, and returns all possible values for the subsets $A_1,...,A_n$.

To be clear, the condition means that $A_{i_0}$ is the union of $A_{j_0},...,A_{j_t}$ and $A_{j_k}\cap A_{j_{k'}}=\emptyset$ for all sets in the RHS.

What's a good way to go about this?

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Example: Take $N=1$ and $a_0=a_1=a_3=a_4=1$ and $a_2=2$, subject to: $A_2=A_0\sqcup A_1$, $A_2=A_3\sqcup A_4$. There are exactly 4 such families of subsets:

• $A_0=A_3=\{0\}$, $A_1=A_4=\{1\}$, $A_2=\{0,1\}$

• $A_0=A_4=\{0\}$, $A_1=A_3=\{1\}$, $A_2=\{0,1\}$

• $A_1=A_3=\{0\}$, $A_0=A_4=\{1\}$, $A_2=\{0,1\}$

• $A_1 =A_4=\{0\}$, $A_0=A_3=\{1\}$, $A_2=\{0,1\}$

Notes: I suspect the answer is some sort of recursive function, but I haven't been able to do this by myself. I also suspect this can be translated into a graph coloring problem. Incidentally, here is the (as of yet unanswered) math version of this question

Answer 1

Given your comments that the numbers involved here are all fairly small, I recommend you use a SAT solver. There is a straightforward encoding: introduce boolean variables $x_{i,\ell}$, with the intended meaning that $x_{i,\ell}$ is true iff $\ell \in A_i$. Then all of your constraints can be translated into CNF clauses, and you can search for a satisfying assignment to the resulting CNF formula using a SAT solver.

The trickiest part will be the cardinality constraints (that $|A_i|=a_i$). That can be encoded using standard methods (see, e.g., Reduce the following problem to SAT, Encoding 1-out-of-n constraint for SAT solvers). In practice, your life will be easier if you use Z3, which provides a convenient front-end to SAT solvers. Z3 supports cardinality constraints natively (see, e.g., https://stackoverflow.com/q/43081929/781723).

Any time you find yourself doing something like recursive search with backtracking etc., consider a SAT solver. At its core, that is more or less what a SAT solver is doing, but folks have worked out powerful and sophisticated methods for doing that intelligently.

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Another approach would be to use an integer linear programming (ILP) solver. In particular, introduce zero-or-one variables $x_{i,\ell}$, with the intended meaning that $x_{i,\ell}=1$ iff $\ell \in A_i$ (the same encoding as above). Now, each constraint can be encoded into a linear inequality. Then, try to solve the resulting problem with an off-the-shelf ILP solver, such as Cplex or Gurobi.

The encoding into ILP is in some sense "simpler" than the encoding into SAT (in particular, the cardinality constraints can be encoded directly in a natural way), so it's possible this might work better.