We introduce and study the single-path design-balanced service network design problem, where flow of a commodity must use only one path from origin to destination. We present an arc-based formulation. We propose two approaches respectively based on matheuristics: local branching and relaxation induced neighborhood search. These approaches hybridize metaheuristic strategies and mathematical programming, and use an idea of defining neighborhoods as small mixed integer programs (MIP), and exploring neighborhoods using MIP solvers. We also present an implementation combining these two approaches. In addition, we propose a Lagrangian heuristic to generate a starting solution. Our computational results demonstrate effectiveness of these approaches.