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Solving parity games, which are equivalent to modal $\mu$-calculus model checking, is a central algorithmic problem in formal methods. Besides the standard computation model with the explicit representation of games, another important…

Computer Science and Game Theory · Computer Science 2019-09-12 Krishnendu Chatterjee , Wolfgang Dvořák , Monika Henzinger , Alexander Svozil

Open parity games are proposed as a compositional extension of parity games with algebraic operations, forming string diagrams of parity games. A potential application of string diagrams of parity games is to describe a large parity game…

Logic in Computer Science · Computer Science 2025-05-29 Kazuki Watanabe

This paper provides a polynomial-time algorithm for solving parity games that runs in $\mathcal{O}(n^{2}\cdot(n + m))$ time-ending a search that has taken decades. Unlike previous attractor-based algorithms, the presented algorithm only…

Data Structures and Algorithms · Computer Science 2025-11-07 Rick van der Heijden

We introduce the novel notion of winning cores in parity games and develop a deterministic polynomial-time under-approximation algorithm for solving parity games based on winning core approximation. Underlying this algorithm are a number…

Computer Science and Game Theory · Computer Science 2016-02-08 Steen Vester

Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we…

Computational Complexity · Computer Science 2009-09-25 Erik D. Demaine , Robert A. Hearn

It is well-known that the winning region of a parity game with $n$ nodes and $k$ priorities can be computed as a $k$-nested fixpoint of a suitable function; straightforward computation of this nested fixpoint requires…

Computational Complexity · Computer Science 2021-03-23 Daniel Hausmann , Lutz Schröder

We present a general framework for solving a large class of learning problems with non-linear functions of classification rates. This includes problems where one wishes to optimize a non-decomposable performance metric such as the F-measure…

Machine Learning · Computer Science 2019-09-09 Harikrishna Narasimhan , Andrew Cotter , Maya Gupta

We consider two-player games played in real time on game structures with clocks and parity objectives. The games are concurrent in that at each turn, both players independently propose a time delay and an action, and the action with the…

Logic in Computer Science · Computer Science 2008-07-09 Krishnendu Chatterjee , Thomas A. Henzinger , Vinayak Prabhu

We analyze different ways of pairing agents in a bipartite matching problem, with regard to its scaling properties and to the distribution of individual ``satisfactions''. Then we explore the role of partial information and bounded…

Statistical Mechanics · Physics 2009-11-10 Paolo Laureti , Yi-Cheng Zhang

We consider two-player games played in real time on game structures with clocks where the objectives of players are described using parity conditions. The games are \emph{concurrent} in that at each turn, both players independently propose…

Logic in Computer Science · Computer Science 2019-03-14 Krishnendu Chatterjee , Thomas A. Henzinger , Vinayak S. Prabhu

Several distinct techniques have been proposed to design quasi-polynomial algorithms for solving parity games since the breakthrough result of Calude, Jain, Khoussainov, Li, and Stephan (2017): play summaries, progress measures and register…

Formal Languages and Automata Theory · Computer Science 2020-01-15 Wojciech Czerwiński , Laure Daviaud , Nathanaël Fijalkow , Marcin Jurdziński , Ranko Lazić , Paweł Parys

Parity games play a central role in model checking and satisfiability checking. Solving parity games is computationally expensive, among others due to the size of the games, which, for model checking problems, can easily contain $10^9$…

Logic in Computer Science · Computer Science 2016-03-22 S. Cranen , J. J. A. Keiren , T. A. C. Willemse

We study strategy improvement algorithms for solving parity games. While these algorithms are known to solve parity games using a very small number of iterations, experimental studies have found that a high step complexity causes them to…

Data Structures and Algorithms · Computer Science 2017-05-08 John Fearnley

We consider the problem of solving random parity games. We prove that parity games exibit a phase transition threshold above $d_P$, so that when the degree of the graph that defines the game has a degree $d > d_P$ then there exists a…

Logic in Computer Science · Computer Science 2020-07-17 Richard Combes , Mikael Touati

Propositional satisfiability (SAT) solvers, which typically operate using conjunctive normal form (CNF), have been successfully applied in many domains. However, in some application areas such as circuit verification, bounded model…

Logic in Computer Science · Computer Science 2013-11-19 Tero Laitinen , Tommi Junttila , Ilkka Niemelä

Classical objectives in two-player zero-sum games played on graphs often deal with limit behaviors of infinite plays: e.g., mean-payoff and total-payoff in the quantitative setting, or parity in the qualitative one (a canonical way to…

Logic in Computer Science · Computer Science 2016-09-21 Véronique Bruyère , Quentin Hautem , Mickael Randour

Matching games naturally generalize assignment games, a well-known class of cooperative games. Interest in matching games has grown recently due to some breakthrough results and new applications. This state-of-the-art survey provides an…

Computer Science and Game Theory · Computer Science 2023-06-22 Márton Benedek , Péter Biró , Matthew Johnson , Daniël Paulusma , Xin Ye

We propose a novel algorithm for the solution of mean-payoff games that merges together two seemingly unrelated concepts introduced in the context of parity games, small progress measures and quasi dominions. We show that the integration of…

Logic in Computer Science · Computer Science 2019-07-16 Massimo Benerecetti , Daniele Dell'Erba , Fabio Mogavero

Infinitely repeated games support equilibrium concepts beyond those present in one-shot games (e.g., cooperation in the prisoner's dilemma). Nonetheless, repeated games fail to capture our real-world intuition for settings with many…

Computer Science and Game Theory · Computer Science 2025-05-22 Henry Fleischmann , Kiriaki Fragkia , Ratip Emin Berker

Many analysis and verifications tasks, such as static program analyses and model-checking for temporal logics reduce to the solution of systems of equations over suitable lattices. Inspired by recent work on lattice-theoretic progress…

Logic in Computer Science · Computer Science 2021-04-20 Paolo Baldan , Barbara König , Tommaso Padoan , Christina Mika-Michalski