The Simplex and Policy-Iteration Methods Are Strongly Polynomial for the Markov Decision Problem with a Fixed Discount Rate

We prove that the classic policy-iteration method [Howard, R. A. 1960. Dynamic Programming and Markov Processes. MIT, Cambridge] and the original simplex method with the most-negative-reduced-cost pivoting rule of Dantzig are strongly polynomial-time algorithms for solving the Markov decision problem (MDP) with a fixed discount rate. Furthermore, the computational complexity of the policy-iteration and simplex methods is superior to that of the only known strongly polynomial-time interior-point algorithm [Ye, Y. 2005. A new complexity result on solving the Markov decision problem. Math. Oper. Res. 30(3) 733-749] for solving this problem. The result is surprising because the simplex method with the same pivoting rule was shown to be exponential for solving a general linear programming problem [Klee, V., G. J. Minty. 1972. How good is the simplex method? Technical report. O. Shisha, ed. Inequalities III. Academic Press, New York], the simplex method with the smallest index pivoting rule was shown to be exponential for solving an MDP regardless of discount rates [Melekopoglou, M., A. Condon. 1994. On the complexity of the policy improvement algorithm for Markov decision processes. INFORMS J. Comput. 6(2) 188-192], and the policy-iteration method was recently shown to be exponential for solving undiscounted MDPs under the average cost criterion. We also extend the result to solving MDPs with transient substochastic transition matrices whose spectral radii are uniformly below one.