Parallel approximation of min-max problems

TL;DR

This paper introduces a parallel approximation algorithm for min-max problems, extending the class of interactions with parallel solutions and proving complexity class collapses, including QIP=PSPACE, through quantum interactive proof simulations.

Contribution

It develops a novel parallel approximation scheme for min-max problems using matrix multiplicative weights, enabling solutions for adaptive two-party quantum interactions and collapsing certain complexity classes.

Findings

First parallel algorithm for adaptive two-party interactions

Collapse of several complexity classes to PSPACE

Polynomial-space simulation of quantum interactive proofs

Abstract

This paper presents an efficient parallel approximation scheme for a new class of min-max problems. The algorithm is derived from the matrix multiplicative weights update method and can be used to find near-optimal strategies for competitive two-party classical or quantum interactions in which a referee exchanges any number of messages with one party followed by any number of additional messages with the other. It considerably extends the class of interactions which admit parallel solutions, demonstrating for the first time the existence of a parallel algorithm for an interaction in which one party 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 specific 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.