A Strongly Polynomial Algorithm for Linear Exchange Markets
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We present a strongly polynomial algorithm for computing an equilibrium in Arrow-Debreu exchange markets with linear utilities. The main measure of progress is identifying a set of edges that must correspond to best bang-per-buck ratios in every equilibrium, called the revealed edge set. We use a variant of the combinatorial algorithm by Duan and Mehlhorn to identify a new revealed edge in a strongly polynomial number of iterations. Every time a new edge is found, we use a subroutine to identify an approximately best possible solution corresponding to the current revealed edge set. Finding the best solution can be reduced to solving a linear program. 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.
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