Towards polynomial simplex-like algorithms for market equilibria

In this paper we consider the problem of computing market equilibria in the Fisher setting for utility models such as spending constraint and perfect, price-discrimination. These models were inspired from modern e-commerce settings and attempt to bridge the gap between the computationally hard but realistic separable, piecewise-linear and concave utility model and, the tractable but less relevant linear utility case. While there are polynomial time algorithms known for these problems, the question of whether there exist polynomial time Simplex-like algorithms has remained elusive, even for linear markets. Such algorithms are desirable due to their conceptual simplicity, ease of implementation and practicality. This paper takes a significant step towards this goal by presenting the first Simplex-like algorithms for these markets assuming a positive resolution of an algebraic problem of Cucker, Koiran and Smale. Unconditionally, our algorithms are FPTASs; they compute prices and allocations such that each buyer derives at least a 1/1+ε-fraction of the utility at a true market equilibrium, and their running times are polynomial in the input length and 1/ε. We start with convex programs which capture market equilibria in each setting and, in a systematic way, convert them into linear complementarity problem (LCP) formulations. Then, departing from previous approaches which try to pivot on a single polyhedron associated to the LCP obtained, we carefully construct a polynomial-length sequence of polyhedra, one containing the other, such that starting from an optimal solution to one allows us to obtain an optimal solution to the next in the sequence in a polynomial number of complementary pivot steps. Our framework to convert a convex program into an LCP and then come up with a Simplex-like algorithm that moves on a sequence of connected polyhedra may be of independent interest.