Stochastic Shortest Paths and Weight-Bounded Properties in Markov Decision Processes

TL;DR

This paper introduces new algorithms for analyzing weighted Markov decision processes, enabling efficient solutions to stochastic shortest path problems and weight-bounded reachability with probabilistic guarantees.

Contribution

It presents novel algorithms for classifying end components and solving classical and qualitative problems in integer-weighted MDPs, extending existing results.

Findings

Polynomial-time algorithm for stochastic shortest path problem.

Decidability results for weight-bounded reachability under probability constraints.

Complexity classifications including NP ∩ coNP and pseudo-polynomial time solutions.

Abstract

The paper deals with finite-state Markov decision processes (MDPs) with integer weights assigned to each state-action pair. New algorithms are presented to classify end components according to their limiting behavior with respect to the accumulated weights. These algorithms are used to provide solutions for two types of fundamental problems for integer-weighted MDPs. First, a polynomial-time algorithm for the classical stochastic shortest path problem is presented, generalizing known results for special classes of weighted MDPs. Second, qualitative probability constraints for weight-bounded (repeated) reachability conditions are addressed. Among others, it is shown that the problem to decide whether a disjunction of weight-bounded reachability conditions holds almost surely under some scheduler belongs to $\textrm{NP}\cap \textrm{coNP}$, is solvable in pseudo-polynomial time and is at least as hard as solving two-player mean-payoff games, while the corresponding problem for universal quantification over schedulers is solvable in polynomial time.