Let $\mathcal{S}$ be a collection of sets. A set straight-line program that enumerates $\mathcal{S}$ is a sequence of sets $B_1,\ldots,B_m$, such that

  • $\mathcal{S}\subseteq \{B_1,\ldots,B_m\}$.
  • For each $i$, either $|B_i|=1$ or $B_i = B_j\cup B_k$ for some $j,k<i$.

Have this kind of problem been studied before?

  • 2
    $\begingroup$ I haven't seen this exactly before, but people have studied circuits over sets of natural numbers, where the operations are $\cap,\cup,+,\times$ and complement, and each gate computes a set of natural numbers. Seems pretty related. See en.wikipedia.org/wiki/Circuits_over_sets_of_natural_numbers and refs therein. $\endgroup$ Apr 11 at 14:41
  • 1
    $\begingroup$ ZDD can be seen as a restricted form of what you want to do (without unions and with an underlying order the elements should go in); OBDD too but the semantics is a bit farther from what you ask. With union, that would give non deterministic OBDD; I do not know whether non det ZDD have been introduced in the literature but that would not be too hard to define (just add $\vee$-gates corresponding to unions). That would give you interesting properties: you can compute size of the set computed by a ZDD, or approximate it with an FPRAS in the non-deterministic case, uniformly sample elements etc. $\endgroup$
    – holf
    Apr 12 at 14:43

1 Answer 1


This problem is basically equivalent to the minimum AND-circuit problem.

They ask for a minimum circuit that only use AND gates, and it tries to compute a set of monomials. A monomial is computed by the circuit if one of the output of some gate is such monomial.

Arpe, Jan; Manthey, Bodo, Approximability of minimum AND-circuits, Algorithmica 53, No. 3, 337-357 (2009). ZBL1172.68061.


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