On the space complexity of linear programming with preprocessing

It is well known that Linear Programming is P-complete, with a log-space reduction. In this work we ask whether Linear Programming remains P-complete, even if the polyhedron (i.e., the set of linear inequality constraints) is a fixed polyhedron, for each input size, and only the objective function is given as input. More formally, we consider the following problem: maximize c x, subject to Ax ≥b; x €2 Rd, where A; b are fixed in advance and only c is given as an input. We start by showing that the problem remains P-complete with a log-space reduction, thus showing that no(1)-space algorithms are unlikely. This result is proved by a direct classical reduction. We then turn to study approximation algorithms and ask what is the best approximation factor that could be obtained by a small space algorithm. Since approximation factors are mostly meaningful when the objective function is nonnegative, we restrict ourselves to the case where x ≥0 and c ≥0. We show that (even in this possibly easier case) approximating the value of max c x (within any polynomial factor) is P-complete with a polylog space reduction, thus showing that 2(log n)o(1)-space approximation algorithms are unlikely. The last result is proved using a recent work of Kalai, Raz, and Rothblum, showing that every language in P has a nosignaling multi-prover interactive proof with poly-logarithmic communication complexity. To the best of our knowledge, our result gives the first space hardness of approximation result proved by a PCP-based argument.