Fixed-Dimensional Energy Games are in Pseudo-Polynomial Time

TL;DR

This paper presents a new algorithm for solving fixed-dimensional energy games efficiently, demonstrating that these problems are solvable in pseudo-polynomial time for dimensions three or higher, and clarifies their computational complexity.

Contribution

It generalizes the hyperplane separation technique to energy games, providing a pseudo-polynomial time algorithm for fixed dimensions and establishing 2EXPTIME-completeness.

Findings

Algorithm runs in exponential time only in the dimension, not in the number of vertices.

Solves the open problem for fixed dimensions ≥3 regarding pseudo-polynomial solvability.

Reduces complexity of multi-dimensional energy games from non-elementary to 2EXPTIME.

Abstract

We generalise the hyperplane separation technique (Chatterjee and Velner, 2013) from multi-dimensional mean-payoff to energy games, and achieve an algorithm for solving the latter whose running time is exponential only in the dimension, but not in the number of vertices of the game graph. This answers an open question whether energy games with arbitrary initial credit can be solved in pseudo-polynomial time for fixed dimensions 3 or larger (Chaloupka, 2013). It also improves the complexity of solving multi-dimensional energy games with given initial credit from non-elementary (Br\'azdil, Jan\v{c}ar, and Ku\v{c}era, 2010) to 2EXPTIME, thus establishing their 2EXPTIME-completeness.