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# Lifted Linear Programming

*Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics*, PMLR 22:788-797, 2012.

#### Abstract

Lifted inference approaches have rendered large, previously intractable probabilistic inference problems quickly solvable by handling whole sets of indistinguishable objects together. Triggered by this success, we show that another important AI technique is liftable, too, namely linear programming. Intuitively, given a linear program (LP), we employ a lifted variant of Gaussian belief propagation (GaBP) to solve the systems of linear equations arising when running an interior-point method to solve the LP. However, this naive solution cannot make use of standard solvers for linear equations and is doomed to construct lifted networks in each iteration of the interior-point method again, an operation that can itself be quite costly. To address both issues, we show how to read off an equivalent LP from the lifted GaBP computations that can be solved using any off-the-shelf LP solver. We prove the correctness of this compilation approach, including a lifted duality theorem, and experimentally demonstrate that it can greatly reduce the cost of solving LPs.