Linear Phase Transition in Random Linear Constraint Satisfaction Problem

Our model is a generalized linear programming relaxation of a much studied random K-SAT problem. Specifically, a set of linear constraints C on K variables is fixed. From a pool of n variables, K variables are chosen uniformly at random and a constraint is chosen from C also uniformly at random. This procedure is repeated m times independently. We ask the following question: is the resulting linear programming problem feasible? We show that the feasibility property experiences a linear phase transition, when n → ∞ and m = cn for some constant c. Namely, there exists a critical value c∗ such that, when c < c∗, the system is feasible or is asymptotically almost feasible, as n → ∞, but, when c > c∗, the ”distance” from feasibility is at least a positive constant independent of n. Our results are obtained using powerful local weak convergence methods developed in Aldous [3], [4], Aldous and Steele [5], Steele [19].