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In this paper, we present a method for solving Mixed Logical/ Integer Linear Programming (MLILP) problems that integrates a polynomial-time ILP solver for the special class of Unit-Two-Variable-Per-Inequality (unit TVPI or UTVPI) constraints of the form ax + by ≤ d, where a, b ∈ {-1, 0, 1}, into generic Boolean SAT solvers. In our approach the linear constraints are viewed as special literals and replaced by binary “indicator” variables to generate a pure logical problem. The resulting problem is subsequently solved using a SAT search procedure which invokes the linear UTVPI solver to incrementally check the consistency of the UTVPI constraints whenever any of the indicator variables are assigned to true. The linear UTVPI solver, on the other hand, can possibly pass implications or no-good constraints to the Boolean SAT solver. Checking the consistency of the UTVPI constraints incrementally enables the UTVPI solver to efficiently interact with the different components of the SAT solver. Additionally, several heuristics and encoding methods are proposed to accommodate the special circumstances of activating UTVPI constraints by the SAT solver. Empirical evidence is presented that demonstrates the advantages of our combined method for large problems.