Approximation Guarantees for Sherali Adams Relaxations

Linear programming (LP) relaxations are widely used as for finding a most likely configuration of a discrete graphical model. Whenever a linear program finds an integral solution it is guaranteed to be optimal and we say the relaxation is tight. In this paper we generalize the Sherali Adams hierarchy of LP relaxations and introduce the use of discrete Fourier Analysis to show this generalization is in a specific sense the best among all LP relaxations as measured by approximation error and characterize some of its structural and geometric properties. We also provide approximation error guarantees for several model reduction techniques such as variable clamping, flippings, and uprootings when coupled with generalized Sherali Adams relaxations. We also introduce and analyze in the same way a novel model reduction technique we call smoothings. This work is primarily theoretical, although we provide several results that can easily be turned into practical algorithms.