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Parameterized complexity of games with monotonically ordered omega-regular objectives

Authors Véronique Bruyère, Quentin Hautem, Jean-François Raskin

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Véronique Bruyère
  • Département d'informatique, Université de Mons (UMONS), Mons, Belgium
Quentin Hautem
  • Département d'informatique, Université de Mons (UMONS), Mons, Belgium
Jean-François Raskin
  • Département d'informatique, Université Libre de Bruxelles (ULB), Brussels, Belgium

Funding

The three authors are supported by COST Action GAMENET CA 16228. Véronique Bruyère and Jean-François Raskin are both supported by the FNRS PDR project "Subgame perfection in graph games" (T.0088.18).
  • Hautem, Quentin: Quentin Hautem is supported by a FRIA fellowship.
  • Raskin, Jean-François: Jean-François Raskin is supported by the ERC Starting Grant inVEST (279499), by the ARC project "Non-Zero Sum Game Graphs: Applications to Reactive Synthesis and Beyond" (Fédération Wallonie-Bruxelles), and by the EOS project "Verifying Learning Artificial Intelligence Systems" (FNRS-FWO), and he is Professeur Francqui de Recherche funded by the Francqui foundation.

Cite AsGet BibTex

Véronique Bruyère, Quentin Hautem, and Jean-François Raskin. Parameterized complexity of games with monotonically ordered omega-regular objectives. In 29th International Conference on Concurrency Theory (CONCUR 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 118, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.CONCUR.2018.29

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 Büchi, coBüchi 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Algorithmic game theory
Keywords
  • two-player zero-sum games played on graphs
  • omega-regular objectives
  • ordered objectives
  • parameterized complexity

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