We present the first combinatorial polynomial time algorithm for computing the equilibrium of the Arrow-Debreu market model with linear utilities. Our algorithm views the allocation of money as flows and iteratively improves the balanced flow as in [Devanur et al. 2008] for Fisher's model. We develop new methods to carefully deal with the flows and surpluses during price adjustments. In our algorithm, we need O(n~6 log(nU)) maximum flow computations, where n is the number of persons and U is the maximum integer utility, and the length of the numbers is at most O(nlog(nU)) to guarantee an exact solution. The previous polynomial time algorithms [Nenakov and Primak 1983, Jain 2007, Ye 2007] for this problem are based on solving convex programs using the ellipsoid algorithm or interior-point method.
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