We show improved NP-hardness of approximating Ordering Constraint Satisfaction Problems (OCSPs). For the two most well-studied OCSPs, MAXIMUM ACYCLIC SUBGRAPH and MAXIMUM BETWEENNESS, we prove NP-hard approximation factors of 14 15 + ε and 1 2 + ε . When it is hard to approximate an OCSP by a constant better than taking a uniformly-at-random ordering, then the OCSP is said to be approximation r...

Given a single network of interactions, asymmetry arises when the links are directed. For example, if protein A upregulates protein B and protein B upregulates protein C, then (in the absence of any further relationships between them) A may affect C but not vice versa. This type of imbalance is reflected in the associated adjacency matrix, which will lack symmetry. A different type of imbalance...

Directed acyclic graphs are commonly used to represent causal relationships among random variables in graphical models. Applications of these models arise in the study of physical and biological systems where directed edges between nodes represent the influence of components of the system on each other. Estimation of directed graphs from observational data is computationally NP-hard. In additio...

The eeciency of a parallel implementation of the conjugate gradient method precon-ditioned by an incomplete Cholesky factorization can vary dramatically depending on the column ordering chosen. One method to minimize the number of major parallel steps is to choose an ordering based on a coloring of the symmetric graph representing the nonzero adjacency structure of the matrix. In this paper, we...

We present a new formulation of the maximum clique problem of a graph in complex space. We start observing that the adjacency matrix A of a graph can always be written in the form A = B where B is a complex, symmetric matrix formed by vectors of zero length (null vectors) and the maximum clique problem can be transformed in a geometrical problem for these vectors. This problem, in turn, is tran...

The problem of community detection receives great attention in recent years. Many methods have been proposed to discover communities in networks. In this paper, we propose a Gaussian stochastic blockmodel that uses Gaussian distributions to fit weight of edges in networks for non-overlapping community detection. The maximum likelihood estimate of this model is equivalent to label propagation wi...

In edge colouring it is often useful to have information about the degree distribution of the neighbours of a given vertex. For example, the well known Vizing’s Adjacency Lemma provides a useful lower bound on the number of vertices of maximum degree adjacent to a given one in a critical graph. We consider an extension of this problem, where we seek information on vertices at distance two from ...

Let m = 2k + 1 ∈ Z be odd. The m-cube graph Hm is the Hasse diagram of the Boolean lattice on the coordinate set Zm. A rooted binary tree T is introduced that has as its nodes the translation classes mod m of the weight-k vertices of all Hm, for 0 < k ∈ Z, with an equivalent form of T whose nodes are the translation classes mod m of weight-(k + 1) vertices via complemented reversals of the form...