# Percentile queries in multi-dimensional Markov decision processes

@article{Randour2015PercentileQI, title={Percentile queries in multi-dimensional Markov decision processes}, author={Mickael Randour and Jean-François Raskin and Ocan Sankur}, journal={Formal Methods in System Design}, year={2015}, volume={50}, pages={207-248} }

Markov decision processes (MDPs) with multi-dimensional weights are useful to analyze systems with multiple objectives that may be conflicting and require the analysis of trade-offs. We study the complexity of percentile queries in such MDPs and give algorithms to synthesize strategies that enforce such constraints. Given a multi-dimensional weighted MDP and a quantitative payoff function f, thresholds $$v_i$$vi (one per dimension), and probability thresholds $$\alpha _i$$αi, we show how to…

## 61 Citations

### L O ] 7 D ec 2 01 6 Percentile Queries in Multi-Dimensional Markov Decision Processes ⋆

- Computer Science, Economics
- 2018

Given a multi-dimensional weighted MDP and a quantitative payoff function f, thresholds vi, and probability thresholds αi, it is shown how to compute a single strategy to enforce that for all dimensions i, the probability of outcomes satisfying fi(ρ) ≥ vi is at least αi.

### Unifying Two Views on Multiple Mean-Payoff Objectives in Markov Decision Processes

- Computer Science2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science
- 2015

This work considers Markov decision processes with multiple limit-average objectives with multiple mean-payoff objectives, and presents a complete characterization of the strategy complexity (in terms of memory bounds and randomization) required to solve the problem.

### Simple Strategies in Multi-Objective MDPs (Technical Report)

- Computer ScienceArXiv
- 2019

It is shown that checking whether a point is achievable by a pure stationary strategy is NP-complete, even for two objectives, and the authors provide an MILP encoding to solve the corresponding problem.

### The Odds of Staying on Budget

- Computer Science, MathematicsICALP
- 2015

This work studies the computational complexity of deciding whether the probability of paths whose accumulated cost satisfies a Boolean combination of inequalities exceeds a given threshold, and shows that this problem is PP-complete, whereas it is hard for the PosSLP problem and in PSpace for general Markov chains.

### Simple Strategies in Multi-Objective MDPs

- Computer ScienceTACAS
- 2020

It is shown that checking whether a point is achievable by a pure stationary strategy is NP-complete, even for two objectives, and the author provides an MILP encoding to solve the corresponding problem.

### Multi-cost Bounded Tradeoff Analysis in MDP

- Computer ScienceJournal of Automated Reasoning
- 2020

The need for more detailed visual presentations of results beyond Pareto curves is discussed and a first visualisation approach that exploits all the available information from the algorithm to support decision makers is presented.

### Interval Markov Decision Processes with Multiple Objectives

- Computer Science, MathematicsACM Trans. Model. Comput. Simul.
- 2019

This article considers Interval Markov decision processes (IMDPs), which generalise classical MDPs by having interval-valued transition probabilities, and investigates the problem of robust multi-objective synthesis for IMDPs and Pareto curve analysis of multi- objective queries on IM DPs and shows that the multi-Objective synthesis problem is PSPACE-hard.

### Multidimensional beyond Worst-Case and Almost-Sure Problems for Mean-Payoff Objectives

- Computer Science2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science
- 2015

The multidimensional BAS threshold problem is solvable in P. This solves the infinite-memory threshold problem left open by Bruyère et al., and this complexity cannot be improved without improving the currently known complexity of classical mean-payoff games.

### Interval Markov Decision Processes with Multiple Objectives: from Robust Strategies to Pareto Curves

- Computer Science
- 2019

This article considers Interval Markov decision processes ( IMDP s), which generalise classical MDP s by having interval-valued transition probabilities and investigates the problem of robust multi-objective synthesis for IMDP and Pareto curve analysis of multi- objective queries on IMDP, and shows that the multi-Objective synthesis problem is PSPACE -hard.

### Computing quantiles in Markov chains with multi-dimensional costs

- Computer Science, Mathematics2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
- 2017

This paper presents an algorithm that allows to approximate the probabilities with arbitrary precision of the total cost, and enables it to show that a decision version of the cost problem lies in the counting hierarchy, a counting analogue to the polynomial-time hierarchy that contains the latter and is included in PSPACE.

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