The GKK Algorithm is the Fastest over Simple Mean-Payoff Games

TL;DR

This paper demonstrates that the GKK algorithm for certain mean-payoff games is faster than previous bounds, with a new analysis reducing its iteration count and connecting it to existing runtime bounds.

Contribution

The paper introduces a symmetric analysis of the GKK algorithm, lowering its iteration bound and relating it to the best known combinatorial runtime bounds.

Findings

Lowered the iteration bound of GKK to N + Ep + Em

Improved the runtime bound from O(mnN) to a structure-aware bound

Connected GKK analysis to the state-of-the-art combinatorial bound of O(m 2^{n/2})

Abstract

We study the algorithm of Gurvich, Khachyian and Karzanov (GKK algorithm) when it is ran over mean-payoff games with no simple cycle of weight zero. We propose a new symmetric analysis, lowering the $O(n^2 N)$ upper-bound of Pisaruk on the number of iterations down to $N + Ep + Em$, which is smaller than $nN$, where $n$ is the number of vertices, $N$ is the largest absolute value of a weight, and $Ep$ and $Em$ are respectively the largest finite energy and dual-energy values of the game. Since each iteration is computed in $O(m)$, this improves on the state of the art pseudopolynomial $O(mnN)$ runtime bound of Brim, Chaloupka, Doyen, Gentilini and Raskin, by taking into account the structure of the game graph. We complement our result by showing that the analysis of Dorfman, Kaplan and Zwick also applies to the GKK algorithm, which is thus also subject to the state of the art combinatorial runtime bound of $O(m 2^{n/2})$.