PPAD-membership for Problems with Exact Rational Solutions: A General Approach via Convex Optimization

TL;DR

This paper presents a novel convex optimization-based technique using a linear-OPT-gate to prove membership of search problems with exact rational solutions in PPAD, simplifying and extending existing proofs across multiple domains.

Contribution

It introduces the linear-OPT-gate as a versatile tool for transforming existence proofs into PPAD-membership proofs, broadening the scope of problems with rational solutions.

Findings

Simplifies existing PPAD-membership proofs in game theory and auctions.

Establishes new PPAD-membership results for various economic problems.

Provides a general framework for rational solutions in total search problems.

Abstract

We introduce a general technique for proving membership of search problems with exact rational solutions in PPAD, one of the most well-known classes containing total search problems with polynomial-time verifiable solutions. In particular, we construct a "pseudogate", coined the linear-OPT-gate, which can be used as a "plug-and-play" component in a piecewise-linear (PL) arithmetic circuit, as an integral component of the "Linear-FIXP" equivalent definition of the class. The linear-OPT-gate can solve several convex optimization programs, including quadratic programs, which often appear organically in the simplest existence proofs for these problems. This effectively transforms existence proofs to PPAD-membership proofs, and consequently establishes the existence of solutions described by rational numbers. Using the linear-OPT-gate, we are able to significantly simplify and generalize almost all known PPAD-membership proofs for finding exact solutions in the application domains of game theory, competitive markets, auto-bidding auctions, and fair division, as well as to obtain new PPAD-membership results for problems in these domains.