Parallel Approximation of Min-max Problems with Applications to Classical and Quantum Zero-Sum Games
2012 IEEE 27th Conference on Computational Complexity
This paper presents an efficient parallel algorithm for a new class of min-max problems based on the matrix multiplicative weights update method. Our algorithm can be used to find near-optimal strategies for competitive two-player classical or quantum games in which a referee exchanges any number of messages with one player followed by any number of additional messages with the other. This algorithm considerably extends the class of games which admit parallel solutions, demonstrating for the
... st time the existence of a parallel algorithm for a game in which one player reacts adaptively to the other. As a consequence, we prove that several competing-provers complexity classes collapse to PSPACE such as QRG(2), SQG and two new classes called DIP and DQIP. A special case of our result is a parallel approximation scheme for a new class of semidefinite programs whose feasible region consists of lists of semidefinite matrices that satisfy a "transcript-like" consistency condition. Applied to this special case, our algorithm yields a direct polynomial-space simulation of multi-message quantum interactive proofs resulting in a first-principles proof of QIP=PSPACE. We also describe parallel implementations of this oracle for certain sets P, yielding an unconditionally efficient parallel approximation algorithm for the min-max problem (2) for those choices of P. Applications to zero-sum games: This algorithm can be used to find near-optimal strategies for a new class of competitive two-player games that are moderated by a referee and obey the following protocol: 1) The referee exchanges several messages only with Alice.