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) over tilde (nmh + (n/h)(3)) time using (O) over tilde (n(2) /h + m) space for any parameter h epsilon [1; n]. In particular, using subquadratic space we get (O) over tilde (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-epsilon)). Moreover, using linear space we match the randomized (O) over tilde (nm + n(3)) time bound of Cohen and Megiddo [SICOMP'94] that required (theta) over tilde (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) over tilde (n(3/2)m(3/4)) time, which improves significantly upon the randomized bound (O) over tilde (n(2) root m) of Madani et al.