Convergence of optimal values past critical threshold

In this paper we examine maximum weight independent sets (MWIS) on branching processes and Erdos-Renyi graphs. Gamarnik, Nowicki, and Swirscsz, showed that, if a particular fixed point equation had a unique solution, then the MWIS value converged. This followed by showing that there was no long range dependencies in the MWIS solution. Here we examine what happens when there are multiple solutions to the fixed point equation, and hence there are long range dependencies. Specifically, we show that, under surprisingly weak conditions - which include the possibility of long range dependencies - that the MWIS value converges in probability to a random variable. We give a new method for calculating the MWIS value for Erdos-Renyi graphs. In the case of vertex weights independently drawn from the standard exponential distribution and with multiple solutions to the fixed point equation, we use our method to show improved upper and lower bounds of the MWIS value. Finally, we show how our proof techniques are general enough to be applied to other optimization problems with local constraints such as maximum weight matching and maximum weight color.