Coupling different integer encodings for SAT

Boolean satisfiability (SAT) solvers have dramatically improved their performance in the last twenty years, enabling them to solve large and complex problems. More recently MaxSAT solvers have appeared that efficiently solve optimisation problems based on SAT. This means that SAT solvers have become a competitive technology for tackling discrete optimisation problems. A challenge in using SAT solvers for discrete optimisation is the many choices of encoding a problem into SAT. When encoding integer variables appearing in discrete optimisation problems, SAT must choose an encoding for each variable. Typical approaches fix a common encoding for all variables. However, different constraints are much more effective when encoded with a particular encoding choice. This inevitably leads to models where variables have different variable encodings. These models must then be able to couple encodings, either by using multiple encoding of single variables and channelling between the representations, or by encoding constraints using a mix of representations for the variables involved. In this paper we show how using mixed encodings of integers and coupled encodings of constraints can lead to better (Max)SAT models of discrete optimisation problems.