Value Iteration with Guessing for Markov Chains and Markov Decision Processes

Two standard models for probabilistic systems are Markov chains (MCs) and Markov decision processes (MDPs). Classic objectives for such probabilistic models for control and planning problems are reachability and stochastic shortest path. The widely studied algorithmic approach for these problems is the Value Iteration (VI) algorithm which iteratively applies local updates called Bellman updates. There are many practical approaches for VI in the literature but they all require exponentially many Bellman updates for MCs in the worst case. A preprocessing step is an algorithm that is discrete, graph-theoretical, and requires linear space. An important open question is whether, after a polynomial-time preprocessing, VI can be achieved with sub-exponentially many Bellman updates. In this work, we present a new approach for VI based on guessing values. Our theoretical contributions are twofold. First, for MCs, we present an almost-linear-time preprocessing algorithm after which, along with guessing values, VI requires only subexponentially many Bellman updates. Second, we present an improved analysis of the speed of convergence of VI for MDPs. Finally, we present a practical algorithm for MDPs based on our new approach. Experimental results show that our approach provides a considerable improvement over existing VI-based approaches on several benchmark examples from the literature.

Linear Equations with Min and Max Operators: Computational Complexity

We consider a class of optimization problems defined by a system of linear equations with min and max operators. This class of optimization problems has been studied under restrictive conditions, such as, (C1) the halting or stability condition; (C2) the non-negative coefficients condition; (C3) the sum up to 1 condition; and (C4) the only min or only max oerator condition. Several seminal results in the literature focus on special cases. For example, turn-based stochastic games correspond to conditions C2 and C3; and Markov decision process to conditions C2, C3, and C4. However, the systematic computational complexity study of all the cases has not been explored, which we address in this work. Some highlights of our results are: with conditions C2 and C4, and with conditions C3 and C4, the problem is NP-complete, whereas with condition C1 only, the problem is in UP intersects coUP. Finally, we establish the computational complexity of the decision problem of checking the respective conditions.