Ising formulations of many NP problems

We provide Ising formulations for many NP-complete and NP-hard problems, including all of Karp's 21 NP-complete problems. This collects and extends mappings to the Ising model from partitioning, covering and satisfiability. In each case, the required number of spins is at most cubic in the size of the problem. This work may be useful in designing adiabatic quantum optimization algorithms.Comment: 27 pages; v2: substantial revision to intro/conclusion, many more references; v3: substantial revision and extension, to-be-published versio

A Lagrangian relaxation approach to the edge-weighted clique problem

The $b$-clique polytope $CP^n_b$ is the convex hull of the node and edge incidence vectors of all subcliques of size at most $b$ of a complete graph on $n$ nodes. Including the Boolean quadric polytope $QP^n$ as a special case and being closely related to the quadratic knapsack polytope, it has received considerable attention in the literature. In particular, the max-cut problem is equivalent with optimizing a linear function over $QP^n_n$. The problem of optimizing linear functions over $CP^n_b$ has so far been approached via heuristic combinatorial algorithms and cutting-plane methods. We study the structure of $CP^n_b$ in further detail and present a new computational approach to the linear optimization problem based on Lucena's suggestion of integrating cutting planes into a Lagrangian relaxation of an integer programming problem. In particular, we show that the separation problem for tree inequalities becomes polynomial in our Lagrangian framework. Finally, computational results are presented. \u

Efficient Relaxations for Dense CRFs with Sparse Higher Order Potentials

Dense conditional random fields (CRFs) have become a popular framework for modelling several problems in computer vision such as stereo correspondence and multi-class semantic segmentation. By modelling long-range interactions, dense CRFs provide a labelling that captures finer detail than their sparse counterparts. Currently, the state-of-the-art algorithm performs mean-field inference using a filter-based method but fails to provide a strong theoretical guarantee on the quality of the solution. A question naturally arises as to whether it is possible to obtain a maximum a posteriori (MAP) estimate of a dense CRF using a principled method. Within this paper, we show that this is indeed possible. We will show that, by using a filter-based method, continuous relaxations of the MAP problem can be optimised efficiently using state-of-the-art algorithms. Specifically, we will solve a quadratic programming (QP) relaxation using the Frank-Wolfe algorithm and a linear programming (LP) relaxation by developing a proximal minimisation framework. By exploiting labelling consistency in the higher-order potentials and utilising the filter-based method, we are able to formulate the above algorithms such that each iteration has a complexity linear in the number of classes and random variables. The presented algorithms can be applied to any labelling problem using a dense CRF with sparse higher-order potentials. In this paper, we use semantic segmentation as an example application as it demonstrates the ability of the algorithm to scale to dense CRFs with large dimensions. We perform experiments on the Pascal dataset to indicate that the presented algorithms are able to attain lower energies than the mean-field inference method