A Strongly Polynomial Algorithm for Linear Exchange Markets

• Published in: Operations research 2023-03, Vol.71 (2), p.487-505
• Main Authors: Garg, Jugal, Végh, László A.
• Format: Article
• Language: English
• Subjects: Algorithms; linear exchange markets; Linear programming; market equilibria; Operations research; Optimization; Polynomials; Solutions; strongly polynomial algorithm; Subroutines; Supply & demand; two-variable-per-inequality systems
• ISSN: 0030-364X; 1526-5463
• DOI: 10.1287/opre.2021.2258

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—that is, pairs of agents and goods that must correspond to the 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. Funding: Financial support from the Division of Computing and Communication Foundations, National Science Foundation (NSF) [Grants 1755619 and 1942321] and from European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme [Grant ScaleOpt-757481] is gratefully acknowledged.