Parameterized complexity of games with monotonically ordered {\omega}-regular objectives

TL;DR

This paper analyzes the parameterized complexity of two-player games with monotonic {5}-regular objectives, providing fixed-parameter tractable algorithms and complexity results for various objectives and preorders.

Contribution

It offers a detailed complexity analysis of threshold problems in monotonic ordered {5}-regular objectives, including polynomial algorithms for specific cases and strategies.

Findings

Threshold problem is in FPT for all monotonic preorders and classical objectives.

Polynomial algorithms developed for Bbchi, coBbchi, and explicit Muller objectives.

Complexity of computing values and strategies analyzed for lexicographic preorder.

Abstract

In recent years, two-player zero-sum games with multiple objectives have received a lot of interest as a model for the synthesis of complex reactive systems. In this framework, Player 1 wins if he can ensure that all objectives are satisfied against any behavior of Player 2. When this is not possible to satisfy all the objectives at once, an alternative is to use some preorder on the objectives according to which subset of objectives Player 1 wants to satisfy. For example, it is often natural to provide more significance to one objective over another, a situation that can be modelled with lexicographically ordered objectives for instance. Inspired by recent work on concurrent games with multiple {\omega}-regular objectives by Bouyer et al., we investigate in detail turned-based games with monotonically ordered and {\omega}-regular objectives. We study the threshold problem which asks whether player 1 can ensure a payoff greater than or equal to a given threshold w.r.t. a given monotonic preorder. As the number of objectives is usually much smaller than the size of the game graph, we provide a parametric complexity analysis and we show that our threshold problem is in FPT for all monotonic preorders and all classical types of {\omega}-regular objectives. We also provide polynomial time algorithms for B\"uchi, coB\"uchi and explicit Muller objectives for a large subclass of monotonic preorders that includes among others the lexicographic preorder. In the particular case of lexicographic preorder, we also study the complexity of computing the values and the memory requirements of optimal strategies.