A Hike in the Phases of the 1-in-3 Satisfiability

We summarise our results for the random $\epsilon$--1-in-3 satisfiability problem, where $\epsilon$ is a probability of negation of the variable. We employ both rigorous and heuristic methods to describe the SAT/UNSAT and Hard/Easy transitions.Comment: 2 pages, introductory level, proceed. for the Les Houches Session LXXXV 2006 on Complex System

MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies

Finding the most probable assignment (MAP) in a general graphical model is known to be NP hard but good approximations have been attained with max-product belief propagation (BP) and its variants. In particular, it is known that using BP on a single-cycle graph or tree reweighted BP on an arbitrary graph will give the MAP solution if the beliefs have no ties. In this paper we extend the setting under which BP can be used to provably extract the MAP. We define Convex BP as BP algorithms based on a convex free energy approximation and show that this class includes ordinary BP with single-cycle, tree reweighted BP and many other BP variants. We show that when there are no ties, fixed-points of convex max-product BP will provably give the MAP solution. We also show that convex sum-product BP at sufficiently small temperatures can be used to solve linear programs that arise from relaxing the MAP problem. Finally, we derive a novel condition that allows us to derive the MAP solution even if some of the convex BP beliefs have ties. In experiments, we show that our theorems allow us to find the MAP in many real-world instances of graphical models where exact inference using junction-tree is impossible

Globally optimal solutions for energy minimization in stereo vision using reweighted belief propagation

Abstract A wide range of low level vision problems have been for-mulated in terms of finding the most probable assignment of a Markov Random Field (or equivalently the lowest en-ergy configuration). Perhaps the most successful example is stereo vision. For the stereo problem, it has been shown thatfinding the global optimum is NP hard but good results have been obtained using a number of approximate optimizationalgorithms