This paper introduces a low complexity and optimum decoding algorithm for tail-biting codes. The algorithm, called Selective Initialization Viterbi Algorithm (SIVA), performs the Viterbi algorithm (VA) iteratively and assigns at each iteration the initial costs for a selective set of states to satisfy a necessary condition. The process of selecting the set of states and setting their initial costs is done by forming a directed acyclic graph among the candidate states. We prove the convergence and optimality of SIVA and analyze its complexity in terms of the number of operations for the worst-case scenario with a noise-like decoder input. SIVA achieves optimum decoding at a complexity comparable to the popular, yet sub-optimal, wrap-around Viterbi algorithm (WAVA) and at several orders of magnitude lower complexity compared to other optimal tail-biting decoding algorithms. Application of SIVA to a practical millimeter wave MIMO system with two different tail-biting codes illustrates URLLC-regime frame error performance better than WAVA and confirms the low complexity of the proposed algorithm.