# Linear phase transition in random linear constraint satisfaction problems

## Abstract

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 are interested in whether the resulting linear programming problem is feasible. We prove that the feasibility property experiences a linear phase transition, when n → ∞ and m = cn for a constant c. Namely, there exists a critical value c* such that, when c < c*, the problem is feasible or is asymptotically almost feasible, as n → ∞, but, when c > c*, the "distance" to feasibility is at least a positive constant independent of n. Our result is obtained using the combination of a powerful local weak convergence method developed in Aldous [Ald92], [Ald01], Aldous and Steele [AS03], Steele [Ste02] and martingale techniques. By exploiting a linear programming duality, our theorem implies the following result in the context of sparse random graphs G(n, cn) on n nodes with cn edges, where edges are equipped with randomly generated weights. Let M (n, c) denote maximum weight matching in G(n, cn). We prove that when c is a constant and n → ∞, the limit limn→∞ ℳ(n, c)/n, exists, with high probability. We further extend this result to maximum weight b-matchings also in G(n, cn).