**Abstract: **Using the powerful machinery of the linear complementarity problem and Lemke's algorithm, we give a practical algorithm for computing an equilibrium for Fisher and Arrow-Debreu markets under separable, piecewise-linear concave (SPLC) utilities, despite the PPAD-completeness of this case.

In 1975, Eaves had given such an algorithm for the case of linear utilities and had asked for an extension to the piecewise-linear, concave utilities. Our result settles the relevant subcase of his problem as well as the problem of Vazirani and Yannakakis of obtaining a path following algorithm for SPLC markets, thereby giving a direct proof of membership of this case in PPAD.

We also prove that SPLC markets have an odd number of equilibria (up to scaling), hence matching the classical result of Shapley (1974), which was based on the Lemke-Howson algorithm and shows a similar fact about Nash equilibria of a 2-person bimatrix game.

For the linear case, Eaves had asked for a combinatorial interpretation of his algorithm. We provide this and use it to get a particularly simple proof of the fact that the set of equilibrium prices is convex.

(This is joint work with Jugal Garg, Ruta Mehta and Milind Sohoni.)