Perfect Half Space Games

TL;DR

This paper introduces perfect half space games, a new class of multi-dimensional games, and demonstrates their application in solving multi-dimensional energy parity games efficiently with fixed dimensions.

Contribution

It establishes the reduction of bounding games to perfect half space games and their translation to lexicographic energy games, enabling pseudo-polynomial solutions for multi-dimensional energy parity games.

Findings

Bounding games reduce to perfect half space games.

Perfect half space games are positionally determined.

Multi-dimensional energy parity games can be solved in pseudo-polynomial time.

Abstract

We introduce perfect half space games, in which the goal of Player 2 is to make the sums of encountered multi-dimensional weights diverge in a direction which is consistent with a chosen sequence of perfect half spaces (chosen dynamically by Player 2). We establish that the bounding games of Jurdzi\'nski et al. (ICALP 2015) can be reduced to perfect half space games, which in turn can be translated to the lexicographic energy games of Colcombet and Niwi\'nski, and are positionally determined in a strong sense (Player 2 can play without knowing the current perfect half space). We finally show how perfect half space games and bounding games can be employed to solve multi-dimensional energy parity games in pseudo-polynomial time when both the numbers of energy dimensions and of priorities are fixed, regardless of whether the initial credit is given as part of the input or existentially quantified. This also yields an optimal 2-EXPTIME complexity with given initial credit, where the best known upper bound was non-elementary.