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Segment routing (SR) combines the advantages of source routing supported by centralized software-defined networking (SDN) paradigm and hop-by-hop routing applied in distributed IP network infrastructure. However, because of the computation inefficiency, it is nearly impossible to evaluate whether various types of networks will benefit from the SR with multiple segments using conventional approaches. In this paper, we propose a flexible $Q$-SR model as well as its formulation in order to fully explore the potential of SR from an algorithmic perspective. The model leads to a highly extensible framework to design and evaluate algorithms that can be adapted to various network topologies and traffic matrices. For the offline setting, we develop a fully polynomial time approximation scheme (FPTAS) which can finds a $(1+ω)$-approximation solution for any specified $ω>0$ in time that is a polynomial function of the network size. To the best of our knowledge, the proposed FPTAS is the first algorithm that can compute arbitrarily accurate solution. For the online setting, we develop an online primal-dual algorithm that proves $O(1)$-competitive and violates link capacities by a factor of $O(\log n)$, where $n$ is the node number. We also prove performance bounds for the proposed algorithms. We conduct simulations on realistic topologies to validate SR parameters and algorithmic parameters in both offline and online scenarios.