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Abstract. We consider feasibility of linear integer programs in the context of verification systems such as SMT solvers or theorem provers. Although satisfiability of linear integer programs is decidable, many stateof-the-art solvers neglect termination in favor of efficiency. It is challenging to design a solver that is both terminating and practically efficient. Recent work by Jovanović and de Moura constitutes an important step into this direction. Their algorithm CUTSAT is sound, but does not terminate, in general. In this paper we extend their CUTSAT algorithm by refined inference rules, a new type of conflicting core, and a dedicated rule application strategy. This leads to our algorithm CUTSAT++, which guarantees termination.

We present two tests that solve linear integer arithmetic constraints. These tests are sound and efficiently find solutions for a large number of problems. While many complete methods search along the problem surface for a solution, these tests use cubes to explore the interior of the problems. The tests are especially efficient for constraints with a large number of integer solutions, e.g., those with infinite lattice width. Inside the SMT-LIB benchmarks, we have found almost one thousand problem instances with infinite lattice width, and we have shown the advantage of our cube tests on these instances by comparing our implementation of the cube test with several state-of-the-art SMT solvers. Our implementation is not only several orders of magnitudes faster, but it also solves all instances, which most SMT solvers do not. Finally, we discovered an additional application for our cube tests: the extraction of equalities implied by a system of linear arithmetic inequalities. This extraction is useful both as a preprocessing step for linear integer constraint solving as well as for the combination of theories by the Nelson-Oppen method.