Some algebraic and geometric computations in PSPACE
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Variance-penalized Markov decision processes
Mathematics of Operations Research
Quadratic programming is in NP
Information Processing Letters
Approximation algorithms for indefinite quadratic programming
Mathematical Programming: Series A and B
Two Views on Multiple Mean-Payoff Objectives in Markov Decision Processes
LICS '11 Proceedings of the 2011 IEEE 26th Annual Symposium on Logic in Computer Science
An O(n2) time algorithm for alternating Büchi games
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Markov decision processes with multiple objectives
STACS'06 Proceedings of the 23rd Annual conference on Theoretical Aspects of Computer Science
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Pareto curves for probabilistic model checking
ATVA'12 Proceedings of the 10th international conference on Automated Technology for Verification and Analysis
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We study the complexity of central controller synthesis problems for finite-state Markov decision processes, where the objective is to optimize both the expected mean-payoff performance of the system and its stability. e argue that the basic theoretical notion of expressing the stability in terms of the variance of the mean-payoff (called global variance in our paper) is not always sufficient, since it ignores possible instabilities on respective runs. For this reason we propose alernative definitions of stability, which we call local and hybrid variance, and which express how rewards on each run deviate from the run's own mean-payoff and from the expected mean-payoff, respectively. We show that a strategy ensuring both the expected mean-payoff and the variance below given bounds requires randomization and memory, under all the above semantics of variance. We then look at the problem of determining whether there is a such a strategy. For the global variance, we show that the problem is in PSPACE, and that the answer can be approximated in pseudo-polynomial time. For the hybrid variance, the analogous decision problem is in NP, and a polynomial-time approximating algorithm also exists. For local variance, we show that the decision problem is in NP. Since the overall performance can be traded for stability (and vice versa), we also present algorithms for approximating the associated Pareto curve in all the three cases. Finally, we study a special case of the decision problems, where we require a given expected mean-payoff together with zero variance. Here we show that the problems can be all solved in polynomial time.