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Whittle index is a generalization of Gittins index that provides veryefficient allocation rules for restless multi-armed bandits. In this work, wedevelop an algorithm to test the indexability and compute the Whittle indicesof any finite-state restless bandit arm. This algorithm works in the discountedand non-discounted cases, and can compute Gittins index. Our algorithm buildson three tools: (1) a careful characterization of Whittle index that allows oneto compute recursively the kth smallest index from the $(k - 1)$th smallest,and to test indexability, (2) the use of the Sherman-Morrison formula to makethis recursive computation efficient, and (3) a sporadic use of the fastestmatrix inversion and multiplication methods to obtain a subcubic complexity. Weshow that an efficient use of the Sherman-Morrison formula leads to analgorithm that computes Whittle index in $(2/3)n^3 + o(n^3)$ arithmeticoperations, where $n$ is the number of states of the arm. The careful use offast matrix multiplication leads to the first subcubic algorithm to computeWhittle or Gittins index: By using the current fastest matrix multiplication,the theoretical complexity of our algorithm is O(n^2.5286 ). We also develop anefficient implementation of our algorithm that can compute indices of Markovchains with several thousands of states in less than a few seconds.

Testing Indexability and Computing Whittle and Gittins Index in Subcubic Time