# Approximation algorithms for an instance of the Monotone circuit satisfiability

I have the following problem. Given a below boolean formula (of the type explained below) containing $$n$$ literals and two parameter $$k$$ and $$l$$, come up with a satisfying assignment of literals such that no more than $$k$$ of them are set to true. ($$l$$ will be explained below.)

An example boolean formula with 4 literals is as follows: $$(x_1 \lor (x_2 \land x_4)) \land (x_2 \lor (x_3 \land x_1)) \land (x_3 \lor (x_4 \land x_1)) \land (x_4 \lor (x_3 \land x_2))$$. Some properties of the kind of formulae I am dealing with:

1. The number of outermost clauses (connected via AND) is same as the number of literals.
2. There is a restriction on size of the innermost AND clauses, they can have at most $$l$$ literals. For example $$l=2$$ in the example above.

I did some literature search to realize that this is called as the "Monotone circuit satisfiability" problem in the complexity parameterization literature. My interest in the problem is largely practical, so I am looking for the best approximation algorithm known for the above problem. Any help / suggestions are much appreciated.

If this is a practical problem, rather than an approximation algorithm, I suggest using an off-the-shelf SAT solver. There are standard ways to encode the constraint that at most $$k$$ variables are set to true, using an at-most-$$k$$-out-of-$$n$$ constraint. For instance, Z3 has good support for this kind of constraint.