The phase transition of the linear inequalities problem

Abstract

One of the most important problems in the polynomial class is checking the satisfiability of systems of linear inequalities over the rationals. In this paper, we investigate the phase-transition behavior of this problem by adopting a methodology which has been proved very successful on NP-complete problems. The methodology is based on the concept of constrainedness, which characterizes an ensemble of randomly generated problems and allows to predict the location of the phase transition in solving such problems. Our work complements and confirms previous results obtained for other polynomial problems. The approach provides a new characterization of the performance of the Phase I of the Simplex algorithm and allows us to predict its behavior on very large instances by exploiting the technique of finite size scaling.