A polyhedral approach to computing border bases

Border bases can be considered to be the natural extension of Gr\"obner bases that have several advantages. Unfortunately, to date the classical border basis algorithm relies on (degree-compatible) term orderings and implicitly on reduced Gr\"obner bases. We adapt the classical border basis algorithm to allow for calculating border bases for arbitrary degree-compatible order ideals, which is \emph{independent} from term orderings. Moreover, the algorithm also supports calculating degree-compatible order ideals with \emph{preference} on contained elements, even though finding a preferred order ideal is NP-hard. Effectively we retain degree-compatibility only to successively extend our computation degree-by-degree. The adaptation is based on our polyhedral characterization: order ideals that support a border basis correspond one-to-one to integral points of the order ideal polytope. This establishes a crucial connection between the ideal and the combinatorial structure of the associated factor spaces

The world of hereditary graph classes viewed through Truemper configurations

In 1982 Truemper gave a theorem that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities. The characterization states that this can be done for a graph G if and only if it can be done for all induced subgraphs of G that are of few speci c types, that we will call Truemper con gurations. Truemper was originally motivated by the problem of obtaining a co-NP characterization of bipartite graphs that are signable to be balanced (i.e. bipartite graphs whose node-node incidence matrices are balanceable matrices). The con gurations that Truemper identi ed in his theorem ended up playing a key role in understanding the structure of several seemingly diverse classes of objects, such as regular matroids, balanceable matrices and perfect graphs. In this survey we view all these classes, and more, through the excluded Truemper con gurations, focusing on the algorithmic consequences, trying to understand what structurally enables e cient recognition and optimization algorithms

Efficient Exact Inference in Planar Ising Models

We give polynomial-time algorithms for the exact computation of lowest-energy (ground) states, worst margin violators, log partition functions, and marginal edge probabilities in certain binary undirected graphical models. Our approach provides an interesting alternative to the well-known graph cut paradigm in that it does not impose any submodularity constraints; instead we require planarity to establish a correspondence with perfect matchings (dimer coverings) in an expanded dual graph. We implement a unified framework while delegating complex but well-understood subproblems (planar embedding, maximum-weight perfect matching) to established algorithms for which efficient implementations are freely available. Unlike graph cut methods, we can perform penalized maximum-likelihood as well as maximum-margin parameter estimation in the associated conditional random fields (CRFs), and employ marginal posterior probabilities as well as maximum a posteriori (MAP) states for prediction. Maximum-margin CRF parameter estimation on image denoising and segmentation problems shows our approach to be efficient and effective. A C++ implementation is available from http://nic.schraudolph.org/isinf/Comment: Fixed a number of bugs in v1; added 10 pages of additional figures, explanations, proofs, and experiment

DeCAF—Discrimination, Comparison, Alignment Tool for 2D PHarmacophores

Comparison of small molecules is a common component of many cheminformatics workflows, including the design of new compounds and libraries as well as side-effect predictions and drug repurposing. Currently, large-scale comparison methods rely mostly on simple fingerprint representation of molecules, which take into account the structural similarities of compounds. Methods that utilize 3D information depend on multiple conformer generation steps, which are computationally expensive and can greatly influence their results. The aim of this study was to augment molecule representation with spatial and physicochemical properties while simultaneously avoiding conformer generation. To achieve this goal, we describe a molecule as an undirected graph in which the nodes correspond to atoms with pharmacophoric properties and the edges of the graph represent the distances between features. This approach combines the benefits of a conformation-free representation of a molecule with additional spatial information. We implemented our approach as an open-source Python module called DeCAF (Discrimination, Comparison, Alignment tool for 2D PHarmacophores), freely available at http://bitbucket.org/marta-sd/decaf. We show DeCAF’s strengths and weaknesses with usage examples and thorough statistical evaluation. Additionally, we show that our method can be manually tweaked to further improve the results for specific tasks. The full dataset on which DeCAF was evaluated and all scripts used to calculate and analyze the results are also provided