We give the first strongly polynomial time algorithm for computing an equilibrium for the linear utilities case of Fisher's market model. We consider a problem with a set $B$ of buyers and a set $G$ of divisible goods. Each buyer $i$ starts with an initial integral allocation $e_i$ of money. The integral utility for buyer $i$ of good $j$ is $U_{ij}$. We first develop a weakly polynomial time algorithm that runs in $O(n^4 log U_{max} + n^3 e_{max})$ time, where $n = |B| + |G|$. We further modify the algorithm so that it runs in $O(n^4 log n)$ time. These algorithms improve upon the previous best running time of $O(n^8 log U_{max} + n^7 log e_{max})$, due to Devanur et al.
@InProceedings{orlin:DagSemProc.10171.2, author = {Orlin, James B.}, title = {{Improved Algorithms for Computing Fisher's Market Clearing Prices}}, booktitle = {Equilibrium Computation}, pages = {1--19}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {10171}, editor = {Edith Elkind and Nimrod Megiddo and Peter Bro Miltersen and Vijay V. Vazirani and Bernahrd von Stengel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.10171.2}, URN = {urn:nbn:de:0030-drops-26720}, doi = {10.4230/DagSemProc.10171.2}, annote = {Keywords: Market equilibrium, Fisher, strongly polynomial} }
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