Phase transitions in integer linear problems

The resolution of 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.