  In the field of Boolean satisfiability problems (SAT), at-most-k constraints,
which suppress the number of true target variables at most k, are often used to
describe objective problems. At-most-k constraints are used not only for
absolutely necessary constraints (hard constraints) but also for challenging
constraints (soft constraints) to search for better solutions. To encode
at-most-k constraints into Boolean expressions, there is a problem that the
number of Boolean expressions basically increases exponentially with the number
of target variables, so at-most-k often has difficulties for a large number of
variables. To solve this problem, this paper proposes a new encoding method of
at-most-k constraints, called approximate-at-most-k, which has totally fewer
Boolean expressions than conventional methods on the one hand. On the other
hand, it has lost completeness, i.e., some Boolean value assignments that
satisfy the original at-most-k are not allowed with approximate-at-most-k;
hence, it is called approximate. Without completeness, we still have potential
benefits by using them only as soft constraints. For example,
approximate-at-most-16 out of 32 variables requires only 15% of a conventional
at-most-k on the literal number and covers 44% of the solution space. Thus,
approximate-at-most-k can become an alternative encoding method for at-most-k,
especially as soft constraints.
