This paper is devoted to non-linear single-path routing problems, which are known to be NP-hard even in the simplest cases. For solving these problems, we propose an algorithm inspired from Game Theory in which individual flows are allowed to independently select their path to minimize their own cost function. We design the cost function of the flows so that the resulting Nash equilibrium of the game provides an efficient approximation of the optimal solution. We establish the convergence of the algorithm and show that every optimal solution is a Nash equilibrium of the game. We also prove that if the objective function is a polynomial of degree d ≥ 1, then the approximation ratio of the algorithm is (2^{1/d} − 1)^{−d}. Experimental results show that the algorithm provides single-path routings with modest relative errors with respect to optimal solutions, while being several orders of magnitude faster than existing techniques.