Belief Propagation for Linear Programming
This work provides a theoretical extension and practical algorithm for solving a broader class of LP problems using belief propagation, which is incremental but offers specific computational improvements for optimization tasks.
The paper generalizes the connection between belief propagation and linear programming, establishing a tight characterization of LP problems that can be solved via MAP inference relaxations, and proposes an efficient iterative annealing BP algorithm, demonstrating its performance on weighted matching problems with added blossom inequalities.
Belief Propagation (BP) is a popular, distributed heuristic for performing MAP computations in Graphical Models. BP can be interpreted, from a variational perspective, as minimizing the Bethe Free Energy (BFE). BP can also be used to solve a special class of Linear Programming (LP) problems. For this class of problems, MAP inference can be stated as an integer LP with an LP relaxation that coincides with minimization of the BFE at ``zero temperature". We generalize these prior results and establish a tight characterization of the LP problems that can be formulated as an equivalent LP relaxation of MAP inference. Moreover, we suggest an efficient, iterative annealing BP algorithm for solving this broader class of LP problems. We demonstrate the algorithm's performance on a set of weighted matching problems by using it as a cutting plane method to solve a sequence of LPs tightened by adding ``blossom'' inequalities.