We establish the existence of scaling limits for several combinatorial optimization models on Erdos-Renyi and sparse random regular graphs. For a variety of models, including maximum independent sets, MAX-CUT, coloring and K-SAT, we prove that the optimal value appropriately rescaled, converges to a limit w.h.p. as the size of the underlying graph diverges to infinity.
For example, as a special case we prove that the size of a largest independent set in these graphs, normalized by the number of nodes converges to a limit w.h.p. thus resolving an open problem.
Our approach is based on developing a simple combinatorial approach to an interpolation method developed recently by Guerra in the context of statistical physics. Among other things, the Guerra's interpolation method was used to prove the existence of the so-called free energy limits for several spin glass models including Viana-Bray and random K-SAT models. Our simpler combinatorial approach allows us to work with the zero temperature case (optimization) directly and extend the approach to many other models.
Additionally, using our approach we establish the large deviations principle for the satisfiability property for constraint satisfaction problems such as coloring, K-SAT and NAE(Not-All-Equal)-K-SAT.