In software validation, regression testing aims to check the absence of regression faults in new releases of a software system. Typically, test cases used in regression testing are executed during a limited amount of time and are selected to check a given set of user requirements. When testing large systems, the number of regression tests grows quickly over the years, and yet the available time slot stays limited. In order to overcome this problem, an approach known as test suite reduction (TSR), has been developed in software engineering to select a smallest subset of test cases, so that each requirement remains covered at least once. However solving the TSR problem is difficult as the underlying optimization problem is NP-hard, but it is also crucial for vendors interested in reducing the time to market of new software releases. In this paper, we address regression testing and TSR with Constraint Programming (CP). More specifically, we propose new CP models to solve TSR that exploit global constraints, namely NValue and GCC. We reuse a set of preprocessing rules to reduce a priori each instance, and we introduce a structure-aware search heuristic. We evaluated our CP models and proposed improvements against existing approaches, including a simple greedy approach and MINTS, the state-of-the-art tool of the software engineering community. Our experiments show that CP outperforms both the greedy approach and MINTS when it is interfaced with MiniSAT, in terms of percentage of reduction and execution time. When MINTS is interfaced with CPLEX, we show that our CP model performs better only on percentage of reduction. Finally, by working closely with validation engineers from Cisco Systems, Norway, we integrated our CP model into an industrial regression testing process.