Maximum a posteriori (MAP) inference in Markov Random Fields (MRFs) is an NP-hard problem, and thus research has focussed on either finding efficiently solvable subclasses (e.g. trees), or approximate algorithms (e.g. Loopy Belief Propagation (BP) and Tree-reweighted (TRW) methods). This paper presents a unifying perspective of these approximate techniques called “Decomposition Methods”. These are methods that decompose the given problem over a graph into tractable subproblems over subgraphs and then employ message passing over these subgraphs to merge the solutions of the subproblems into a global solution. This provides a new way of thinking about BP and TRW as successive steps in a hierarchy of decomposition methods. Using this framework, we take a principled first step towards extending this hierarchy beyond trees. We leverage a new class of graphs amenable to exact inference, called outerplanar graphs, and propose an approximate inference algorithm called Outer-Planar Decomposition (OPD). OPD is a strict generalization of BP and TRW, and contains both of them as special cases. Our experiments show that this extension beyond trees is indeed very powerful – OPD outperforms current state-of-art inference methods on hard non-submodular synthetic problems and is competitive on real computer vision applications.

We describe a new variational lower-bound on the minimum energy configuration of a planar binary Markov Random Field (MRF). Our method is based on adding auxiliary nodes to every face of a planar embedding of the graph in order to capture the effect of unary potentials. A ground state of the resulting approximation can be computed efficiently by reduction to minimum-weight perfect matching. We show that optimization of variational parameters achieves the same lower-bound as dual-decomposition into the set of all cycles of the original graph. We demonstrate that our variational optimization converges quickly and provides highquality solutions to hard combinatorial problems 10-100x faster than competing algorithms that optimize the same bound. 1

This paper deals with Dynamic MAP Inference, where the goal is to solve an instance of the MAP problem given that we have already solved a related instance of the problem. We propose an algorithm for Dynamic MAP Inference in planar Ising models, called Dynamic Planar-Cuts. As an application of our proposed approach, we show that we can extend the MAP inference algorithm of Schraudolph and Kamenetsky [14] to efficiently compute min-marginals for all variables in the same time complexity as the MAP inference algorithm, which is an O(n) speedup over a naïve approach. 1

We describe a new technique for computing lower-bounds on the minimum energy configuration of a planar Markov Random Field (MRF). Our method successively adds large numbers of constraints and enforces consistency over binary projections of the original problem state space. These constraints are represented in terms of subproblems in a dual-decomposition framework that is optimized using subgradient techniques. The complete set of constraints we consider enforces cycle consistency over the original graph. In practice we find that the method converges quickly on most problems with the addition of a few subproblems and outperforms existing methods for some interesting classes of hard potentials. 1

Contents Quantum Error Correction and fault-tolerant quantum computation represent arguably the most vital theoretical aspect of quantum information processing. It was well known from the early developments of this exciting field that the fragility of coherent quantum systems would be a catastrophic obstacle to the development of large scale quantum computers. The introduction of

Abstract. We describe a new optimization scheme for finding highquality clusterings in planar graphs that uses weighted perfect matching as a subroutine. Our method provides lower-bounds on the energy of the optimal correlation clustering that are typically fast to compute and tight in practice. We demonstrate our algorithm on the problem of image segmentation where this approach outperforms existing global optimization techniques in minimizing the objective and is competitive with the state of the art in producing high-quality segmentations. 1 1