The Complexity of All-switches Strategy Improvement

Strategy improvement is a widely-used and well-studied class of algorithms for solving graph-based infinite games. These algorithms are parameterized by a switching rule, and one of the most natural rules is "all switches" which switches as many edges as possible in each iteration. Continuing a recent line of work, we study all-switches strategy improvement from the perspective of computational complexity. We consider two natural decision problems, both of which have as input a game $G$, a starting strategy $s$, and an edge $e$. The problems are: 1.) The edge switch problem, namely, is the edge $e$ ever switched by all-switches strategy improvement when it is started from $s$ on game $G$? 2.) The optimal strategy problem, namely, is the edge $e$ used in the final strategy that is found by strategy improvement when it is started from $s$ on game $G$? We show $\mathtt{PSPACE}$-completeness of the edge switch problem and optimal strategy problem for the following settings: Parity games with the discrete strategy improvement algorithm of V\"oge and Jurdzi\'nski; mean-payoff games with the gain-bias algorithm [14,37]; and discounted-payoff games and simple stochastic games with their standard strategy improvement algorithms. We also show $\mathtt{PSPACE}$-completeness of an analogous problem to edge switch for the bottom-antipodal algorithm for finding the sink of an Acyclic Unique Sink Orientation on a cube

Symmetric Strategy Improvement

Symmetry is inherent in the definition of most of the two-player zero-sum games, including parity, mean-payoff, and discounted-payoff games. It is therefore quite surprising that no symmetric analysis techniques for these games exist. We develop a novel symmetric strategy improvement algorithm where, in each iteration, the strategies of both players are improved simultaneously. We show that symmetric strategy improvement defies Friedmann's traps, which shook the belief in the potential of classic strategy improvement to be polynomial

An Exponential Lower Bound for the Latest Deterministic Strategy Iteration Algorithms

This paper presents a new exponential lower bound for the two most popular deterministic variants of the strategy improvement algorithms for solving parity, mean payoff, discounted payoff and simple stochastic games. The first variant improves every node in each step maximizing the current valuation locally, whereas the second variant computes the globally optimal improvement in each step. We outline families of games on which both variants require exponentially many strategy iterations

Local Strategy Improvement for Parity Game Solving

The problem of solving a parity game is at the core of many problems in model checking, satisfiability checking and program synthesis. Some of the best algorithms for solving parity game are strategy improvement algorithms. These are global in nature since they require the entire parity game to be present at the beginning. This is a distinct disadvantage because in many applications one only needs to know which winning region a particular node belongs to, and a witnessing winning strategy may cover only a fractional part of the entire game graph. We present a local strategy improvement algorithm which explores the game graph on-the-fly whilst performing the improvement steps. We also compare it empirically with existing global strategy improvement algorithms and the currently only other local algorithm for solving parity games. It turns out that local strategy improvement can outperform these others by several orders of magnitude

Non-oblivious Strategy Improvement

We study strategy improvement algorithms for mean-payoff and parity games. We describe a structural property of these games, and we show that these structures can affect the behaviour of strategy improvement. We show how awareness of these structures can be used to accelerate strategy improvement algorithms. We call our algorithms non-oblivious because they remember properties of the game that they have discovered in previous iterations. We show that non-oblivious strategy improvement algorithms perform well on examples that are known to be hard for oblivious strategy improvement. Hence, we argue that previous strategy improvement algorithms fail because they ignore the structural properties of the game that they are solving

Synthesising Strategy Improvement and Recursive Algorithms for Solving 2.5 Player Parity Games

2.5 player parity games combine the challenges posed by 2.5 player reachability games and the qualitative analysis of parity games. These two types of problems are best approached with different types of algorithms: strategy improvement algorithms for 2.5 player reachability games and recursive algorithms for the qualitative analysis of parity games. We present a method that - in contrast to existing techniques - tackles both aspects with the best suited approach and works exclusively on the 2.5 player game itself. The resulting technique is powerful enough to handle games with several million states

Benchmarks for Parity Games (extended version)

We propose a benchmark suite for parity games that includes all benchmarks that have been used in the literature, and make it available online. We give an overview of the parity games, including a description of how they have been generated. We also describe structural properties of parity games, and using these properties we show that our benchmarks are representative. With this work we provide a starting point for further experimentation with parity games.Comment: The corresponding tool and benchmarks are available from https://github.com/jkeiren/paritygame-generator. This is an extended version of the paper that has been accepted for FSEN 201