A Strongly Polynomial Algorithm for Linear Exchange Markets

TL;DR

This paper introduces a strongly polynomial algorithm for finding market equilibria in linear exchange markets, improving computational efficiency by combining a variant of the DM algorithm with LP approximations.

Contribution

It develops a novel strongly polynomial algorithm for linear exchange markets by leveraging revealed edges and LP approximations, extending previous weakly polynomial methods.

Findings

The algorithm computes market equilibria efficiently.

It uses LP approximations to handle demand-supply constraints.

The approach is strongly polynomial, unlike previous methods.

Abstract

We present a strongly polynomial algorithm for computing an equilibrium in Arrow-Debreu exchange markets with linear utilities. Our algorithm is based on a variant of the weakly-polynomial Duan-Mehlhorn (DM) algorithm. We use the DM algorithm as a subroutine to identify revealed edges, i.e. pairs of agents and goods that must correspond to best bang-per-buck transactions in every equilibrium solution. Every time a new revealed edge is found, we use another subroutine that decides if there is an optimal solution using the current set of revealed edges, or if none exists, finds the solution that approximately minimizes the violation of the demand and supply constraints. This task can be reduced to solving a linear program (LP). Even though we are unable to solve this LP in strongly polynomial time, we show that it can be approximated by a simpler LP with two variables per inequality that is solvable in strongly polynomial time.