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We consider markets in the classical Arrow-Debreumodel. There are n agents and m goods. Each buyer hasa concave utility function (of the bundle of goods he/shebuys) and an initial bundle. At an “equilibrium” set ofprices for goods, if each individual buyer separately ex-changes the initial bundle for an optimal bundle at the setprices, the market clears, i.e., all goods are exactly con-sumed. Classical theorems guarantee the existence of equi-libria, but computing them has been the subject of muchrecent research. In the related area of Multi-Agent Games,much attention has been paid to the complexity as well asalgorithms. While most general problems are hard, poly-nomial time algorithms have been developed for restrictedclasses of games, when one assumes the number of strate-gies is constant.For the Market Equilibrium problem, several importantspecial cases of utility functions have been tackled. Here webegin a program for this problem similar to that for multi-agent games, where general utilities are considered. We be-gin by showing that if the utilities are separable piece-wiselinear concave (PLC) functions, and the number of goods(or alternatively the number of buyers) is constant, thenwe can compute an exact equilibrium in polynomial time.Our technique for the constant number of goods is to de-compose the space of price vectors into cells using certainhyperplanes, so that in each cell, each buyer’s thresholdmarginal utility is known. Still, one needs to solve a linear optimization problem in each cell. We then show the mainresult - that for general (non-separable) PLC utilities, anexact equilibrium can be found in polynomial time providedthe number of goods is constant. The starting point of thealgorithm is a “cell-decomposition” of the space of pricevectors using polynomial surfaces (instead of hyperplanes).We use results from computational algebraic geometry tobound the number of such cells. For solving the probleminside each cell, we introduce and use a novel LP-dualitybased method. We note that if the number of buyers andagents both can vary, the problem is PPAD hard even forthe very special case of PLC utilities - namely Leontief utilities.