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We consider the Arrow–Debreu exchange market model under the assumption that the agents’ demands satisfy the weak gross substitutes (WGS) property. We present a simple auction algorithm that obtains an approximate market equilibrium for WGS demands assuming the availability of a price update oracle. We exhibit specific implementations of such an oracle for WGS demands with bounded price elasticities and for Gale demand systems. 

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. 

The Nash social welfare problem asks for an allocation of indivisible items to agents in order to maximize the geometric mean of agents' valuations. We give an overview of the constant-factor approximation algorithm for the problem when agents have Rado valuations [Garg et al. 2021]. Rado valuations are a common generalization of the assignment (OXS) valuations and weighted matroid rank functions. Our approach also gives the first constant-factor approximation algorithm for the asymmetric Nash social welfare problem under the same valuations, provided that the maximum ratio between the weights is bounded by a constant.