Georgia Institute of Technology College of Sciences

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 with high probability (w.h.p.), as the size of the underlying graph divergesto 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 in the statistical physics literature. Among other things, theinterpolation 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 deviationsprinciple for the satisfiability property for constraint satisfaction problems such as coloring, K-SAT and NAE(Not-All-Equal)-K-SAT. The talk will be completely self-contained. No background on random graph theory/statistical physics is necessary. Joint work with Mohsen Bayati and Prasad Tetali