# Hyperplane separation technique for multidimensional mean-payoff games

Chatterjee K, Velner Y. 2017. Hyperplane separation technique for multidimensional mean-payoff games. Journal of Computer and System Sciences. 88, 236–259.

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Abstract
We consider finite-state and recursive game graphs with multidimensional mean-payoff objectives. In recursive games two types of strategies are relevant: global strategies and modular strategies. Our contributions are: (1) We show that finite-state multidimensional mean-payoff games can be solved in polynomial time if the number of dimensions and the maximal absolute value of weights are fixed; whereas for arbitrary dimensions the problem is coNP-complete. (2) We show that one-player recursive games with multidimensional mean-payoff objectives can be solved in polynomial time. Both above algorithms are based on hyperplane separation technique. (3) For recursive games we show that under modular strategies the multidimensional problem is undecidable. We show that if the number of modules, exits, and the maximal absolute value of the weights are fixed, then one-dimensional recursive mean-payoff games under modular strategies can be solved in polynomial time, whereas for unbounded number of exits or modules the problem is NP-hard.
Publishing Year
Date Published
2017-09-01
Journal Title
Journal of Computer and System Sciences
Acknowledgement
The research was supported by Austrian Science Fund (FWF) Grant No. P 23499-N23, FWF NFN Grant No. S11407-N23 (RiSE), ERC Start grant (279307: Graph Games), Microsoft faculty fellows award, the RICH Model Toolkit (ICT COST Action IC0901), and was carried out in partial fulfillment of the requirements for the Ph.D. degree of the second author.
Volume
88
Page
236 - 259
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Chatterjee K, Velner Y. Hyperplane separation technique for multidimensional mean-payoff games. Journal of Computer and System Sciences. 2017;88:236-259. doi:10.1016/j.jcss.2017.04.005
Chatterjee, K., & Velner, Y. (2017). Hyperplane separation technique for multidimensional mean-payoff games. Journal of Computer and System Sciences. Academic Press. https://doi.org/10.1016/j.jcss.2017.04.005
Chatterjee, Krishnendu, and Yaron Velner. “Hyperplane Separation Technique for Multidimensional Mean-Payoff Games.” Journal of Computer and System Sciences. Academic Press, 2017. https://doi.org/10.1016/j.jcss.2017.04.005.
K. Chatterjee and Y. Velner, “Hyperplane separation technique for multidimensional mean-payoff games,” Journal of Computer and System Sciences, vol. 88. Academic Press, pp. 236–259, 2017.
Chatterjee K, Velner Y. 2017. Hyperplane separation technique for multidimensional mean-payoff games. Journal of Computer and System Sciences. 88, 236–259.
Chatterjee, Krishnendu, and Yaron Velner. “Hyperplane Separation Technique for Multidimensional Mean-Payoff Games.” Journal of Computer and System Sciences, vol. 88, Academic Press, 2017, pp. 236–59, doi:10.1016/j.jcss.2017.04.005.
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