Faster Algorithms for Quantitative Analysis of Markov Chains and Markov Decision Processes with Small Treewidth

TL;DR

This paper introduces faster algorithms for computing key quantitative objectives in Markov Chains and Markov Decision Processes by leveraging small treewidth, significantly improving efficiency on low-treewidth models through theoretical analysis and experimental validation.

Contribution

It presents novel parameterized algorithms that utilize treewidth to accelerate the computation of quantitative objectives in MCs and MDPs, with proven theoretical bounds and practical performance gains.

Findings

Algorithms run in O((n+m)·t^2) time for MCs with treewidth t.

On low-treewidth models, the methods outperform existing algorithms by orders of magnitude.

Experimental results confirm efficiency improvements on benchmark models.

Abstract

Discrete-time Markov Chains (MCs) and Markov Decision Processes (MDPs) are two standard formalisms in system analysis. Their main associated quantitative objectives are hitting probabilities, discounted sum, and mean payoff. Although there are many techniques for computing these objectives in general MCs/MDPs, they have not been thoroughly studied in terms of parameterized algorithms, particularly when treewidth is used as the parameter. This is in sharp contrast to qualitative objectives for MCs, MDPs and graph games, for which treewidth-based algorithms yield significant complexity improvements. In this work, we show that treewidth can also be used to obtain faster algorithms for the quantitative problems. For an MC with $n$ states and $m$ transitions, we show that each of the classical quantitative objectives can be computed in $O((n+m)\cdot t^2)$ time, given a tree decomposition of the MC that has width $t$. Our results also imply a bound of $O(\kappa\cdot (n+m)\cdot t^2)$ for each objective on MDPs, where $\kappa$ is the number of strategy-iteration refinements required for the given input and objective. Finally, we make an experimental evaluation of our new algorithms on low-treewidth MCs and MDPs obtained from the DaCapo benchmark suite. Our experimental results show that on MCs and MDPs with small treewidth, our algorithms outperform existing well-established methods by one or more orders of magnitude.