Exact SAT Solving for the Two-Dimensional Bandwidth Minimization Problem

Abstract

The two-dimensional bandwidth minimization problem (2DBMP) seeks an injective embedding of a guest graph into a square grid that minimizes the maximum Manhattan distance over its edges. Heuristic methods can provide strong upper bounds, but these bounds do not by themselves certify optimality. We present an efficient exact SAT-based approach for 2DBMP that incrementally searches for the minimum feasible bandwidth and certifies optimality through satisfiability and unsatisfiability results. On the standard $\lceil\sqrt n\rceil \times \lceil\sqrt n\rceil$ host grid, under a 3600 s time limit, the proposed SAT approach certifies optimal bandwidths for 41 of 43 Regular instances and 42 of 93 Harwell--Boeing instances, achieving substantially broader optimality certification within the 3600 s time limit than a previous exact approach evaluated with a 72-hour time limit. In addition, it certifies three bandwidth values that improve all previously published comparison values considered in this study and establishes all three as optimal. We further evaluate the approach on alternative host geometries, namely $2\times\lceil n/2\rceil$ and $n\times n$ grids, to assess its effectiveness beyond the standard host. Overall, the results demonstrate that the proposed SAT approach provides an effective exact method for the small- and medium-sized benchmark instances considered in this study, with fewer than 400 vertices, while heuristic methods remain important for larger and more challenging instances.