Correlation Decay in Random Decision Networks

Abstract

We consider a decision network on an undirected graph in which each node corresponds to a decision variable, and each node and edge of the graph is associated with a reward function whose value depends only on the variables of the corresponding nodes. The goal is to construct a decision vector that maximizes the total reward. This decision problem encompasses a variety of models, including maximum-likelihood inference in graphical models (Markov Random Fields), combinatorial optimization on graphs, economic team theory, and statistical physics. The network is endowed with a probabilistic structure in which rewards are sampled from a distribution. Our aim is to identify sufficient conditions on the network structure and rewards distributions to guarantee average-case polynomiality of the underlying optimization problem. Additionally, we wish to characterize the efficiency of a decentralized solution generated on the basis of local information. We construct a new decentralized algorithm called Cavity Expansion and establish its theoretical performance for a variety of graph models and reward function distributions. Specifically, for certain classes of models we prove that our algorithm is able to find a near-optimal solution with high probability in a decentralized way. The success of the algorithm is based on the network exhibiting a certain correlation decay (long-range independence) property, and we prove that this property is indeed exhibited by the models of interest. Our results have the following surprising implications in the area of average-case complexity of algorithms. Finding the largest independent (stable) set of a graph is a well known NP-hard optimization problem for which no polynomial time approximation scheme is possible even for graphs with largest connectivity equal to three unless P D NP. Yet we show that the closely related Maximum Weight Independent Set problem for the same class of graphs admits a PTAS when the weights are independently and identically distributed with the exponential distribution. Namely, randomization of the reward function turns an NP-hard problem into a tractable one. Keywords: optimization; NP-hardness; long-range independence