Improved Strongly Polynomial Algorithms for Deterministic MDPs, 2VPI Feasibility, and Discounted All-Pairs Shortest Paths

We revisit the problem of finding optimal strategies for deterministic Markov Decision Processes (DMDPs), and a closely related problem of testing feasibility of systems of m linear inequalities on n real variables with at most two variables per inequality (2VPI).

We give a randomized trade-off algorithm solving both problems and running in O(nmh + (n/h) 3 ) time using O(n 2 /h + m) space for any parameter h ∈ [1, n]. In particular, using subquadratic space we get O(nm + n 3/2 m 3/4 ) running time, which improves by a polynomial factor upon all the known upper bounds for non-dense instances with m = O(n 2-ǫ ). Moreover, using linear space we match the randomized O(nm + n 3 ) time bound of Cohen and Megiddo [SICOMP'94] that required Θ(n 2 + m) space.

Additionally, we show a new algorithm for the Discounted All-Pairs Shortest Paths problem, introduced by Madani et al. [TALG'10], that extends the DMDPs with optional end vertices. For the case of uniform discount factors, we give a deterministic algorithm running in O(n 3/2 m 3/4 ) time, which improves significantly upon the randomized bound O(n 2 √ m) of Madani et al.