Discounted deterministic Markov decision processes and discounted all-pairs shortest paths

We present two new algorithms for finding optimal strategies for discounted, infinite-horizon, Deterministic Markov Decision Processes (DMDP). The first one is an adaptation of an algorithm of Young, Tarjan and Orlin for finding minimum mean weight cycles. It runs in O(mn + n2 log n) time, where n is the number of vertices (or states) and m is the number of edges (or actions). The second one is an adaptation of a classical algorithm of Karp for finding minimum mean weight cycles. It runs in O(mn) time. The first algorithm has a slightly slower worst-case complexity, but is faster than the first algorithm in many situations. Both algorithms improve on a recent O(mn2)-time algorithm of Andersson and Vorobyov. We also present a randomized Õ(m1/2n2)-time algorithm for finding Discounted All-Pairs Shortest Paths (DAPSP), improving several previous algorithms.