Phase transitions in integer linear problems

The resolution of a linear system with positive integer variables is a basic yet difficult computational problem with many applications. We consider sparse uncorrelated random systems parametrised by the density c and the ratio alpha = N/M between number of variables N and number of constraints M. By means of ensemble calculations we show that the space of feasible solutions endows a Van-Der-Waals phase diagram in the plane (c, alpha). We give numerical evidence that the associated computational problems become more difficult across the critical point and in particular in the coexistence region.