Approximate-At-Most-k Encoding of SAT for Soft Constraints

TL;DR

This paper introduces an approximate encoding method for at-most-k constraints in SAT problems, significantly reducing complexity at the cost of some completeness, making it suitable for soft constraints in large-scale problems.

Contribution

It proposes a novel approximate-at-most-k encoding that reduces Boolean expression size and complexity, addressing scalability issues in SAT formulations.

Findings

Requires only 15% of literals compared to conventional methods

Covers 44% of the solution space despite approximation

Reduces encoding complexity for large at-most-k constraints

Abstract

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.