Algorithms for Maximum Independent Set in Convex Bipartite Graphs

A bipartite graph G = (V, W, E) is convex if there exists an ordering of the vertices of W such that, for each v. V, the neighbors of v are consecutive in W. We describe both a sequential and a BSP/CGM algorithm to find a maximum independent set in a convex bipartite graph. The sequential algorithm improves over the running time of the previously known algorithm and the BSP/CGM algorithm is a parallel version of the sequential one. The complexity of the algorithms does not depend on |W|.

Packing triangles in low degree graphs and indifference graphs

We consider the problems of finding the maximum number of vertex-disjoint triangles (VTP) and edge-disjoint triangles (ETP) in a simple graph. Both problems are NP-hard. The algorithm with the best approximation ratio known so far for these problems has ratio 3/2 + epsilon, a result that follows from a more general algorithm for set packing obtained by Hurkens and Schrijver [On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems, SIAM J. Discrete Math. 2(1) (1989) 68-72]. We present improvements on the approximation ratio for restricted cases of VTP and ETP that are known to be APX-hard: we give an approximation algorithm for VTP on graphs with maximum degree 4 with ratio slightly less than 1.2, and for ETP on graphs with maximum degree 5 with ratio 4/3. We also present an exact linear-time algorithm for VTP on the class of indifference graphs. (C) 2007 Elsevier B.V. All rights reserved.

Sparse partition universal graphs for graphs of bounded degree

In 1983, Chvatal, Trotter and the two senior authors proved that for any Delta there exists a constant B such that, for any n, any 2-colouring of the edges of the complete graph K(N) with N >= Bn vertices yields a monochromatic copy of any graph H that has n vertices and maximum degree Delta. We prove that the complete graph may be replaced by a sparser graph G that has N vertices and O(N(2-1/Delta)log(1/Delta)N) edges, with N = [B`n] for some constant B` that depends only on Delta. Consequently, the so-called size-Ramsey number of any H with n vertices and maximum degree Delta is O(n(2-1/Delta)log(1/Delta)n) Our approach is based on random graphs; in fact, we show that the classical Erdos-Renyi random graph with the numerical parameters above satisfies a stronger partition property with high probability, namely, that any 2-colouring of its edges contains a monochromatic universal graph for the class of graphs on n vertices and maximum degree Delta. The main tool in our proof is the regularity method, adapted to a suitable sparse setting. The novel ingredient developed here is an embedding strategy that allows one to embed bounded degree graphs of linear order in certain pseudorandom graphs. Crucial to our proof is the fact that regularity is typically inherited at a scale that is much finer than the scale at which it is assumed. (C) 2011 Elsevier Inc. All rights reserved.

Geodesics and Almost Geodesic Cycles in Random Regular Graphs

A geodesic in a graph G is a shortest path between two vertices of G. For a specific function e(n) of n, we define an almost geodesic cycle C in G to be a cycle in which for every two vertices u and v in C, the distance d(G)(u, v) is at least d(C)(u, v) - e(n). Let omega(n) be any function tending to infinity with n. We consider a random d-regular graph on n vertices. We show that almost all pairs of vertices belong to an almost geodesic cycle C with e(n)= log(d-1)log(d-1) n+omega(n) and vertical bar C vertical bar =2 log(d-1) n+O(omega(n)). Along the way, we obtain results on near-geodesic paths. We also give the limiting distribution of the number of geodesics between two random vertices in this random graph. (C) 2010 Wiley Periodicals, Inc. J Graph Theory 66: 115-136, 2011

Asymmetric Ramsey Properties of Random Graphs Involving Cliques

Consider the following problem: Forgiven graphs G and F(1),..., F(k), find a coloring of the edges of G with k colors such that G does not contain F; in color i. Rodl and Rucinski studied this problem for the random graph G,,, in the symmetric case when k is fixed and F(1) = ... = F(k) = F. They proved that such a coloring exists asymptotically almost surely (a.a.s.) provided that p <= bn(-beta) for some constants b = b(F,k) and beta = beta(F). This result is essentially best possible because for p >= Bn(-beta), where B = B(F, k) is a large constant, such an edge-coloring does not exist. Kohayakawa and Kreuter conjectured a threshold function n(-beta(F1,..., Fk)) for arbitrary F(1), ..., F(k). In this article we address the case when F(1),..., F(k) are cliques of different sizes and propose an algorithm that a.a.s. finds a valid k-edge-coloring of G(n,p) with p <= bn(-beta) for some constant b = b(F(1),..., F(k)), where beta = beta(F(1),..., F(k)) as conjectured. With a few exceptions, this algorithm also works in the general symmetric case. We also show that there exists a constant B = B(F,,..., Fk) such that for p >= Bn(-beta) the random graph G(n,p) a.a.s. does not have a valid k-edge-coloring provided the so-called KLR-conjecture holds. (C) 2008 Wiley Periodicals, Inc. Random Struct. Alg., 34, 419-453, 2009

The disk-percolation model on graphs

We study a long-range percolation model whose dynamics describe the spreading of an infection on an infinite graph. We obtain a sufficient condition for phase transition and prove all upper bound for the critical parameter of spherically symmetric trees. (C) 2008 Elsevier B.V. All rights reserved.

Extracting Causal Graphs from an Open Provenance Data Model

The open provenance architecture (OPA) approach to the challenge was distinct in several regards. In particular, it is based on an open, well-defined data model and architecture, allowing different components of the challenge workflow to independently record documentation, and for the workflow to be executed in any environment. Another noticeable feature is that we distinguish between the data recorded about what has occurred, emphprocess documentation, and the emphprovenance of a data item, which is all that caused the data item to be as it is and is obtained as the result of a query over process documentation. This distinction allows us to tailor the system to separately best address the requirements of recording and querying documentation. Other notable features include the explicit recording of causal relationships between both events and data items, an interaction-based world model, intensional definition of data items in queries rather than relying on explicit naming mechanisms, and emphstyling of documentation to support non-functional application requirements such as reducing storage costs or ensuring privacy of data. In this paper we describe how each of these features aid us in answering the challenge provenance queries.

An investigation of Hebbian phase sequences as assembly graphs

Hebb proposed that synapses between neurons that fire synchronously are strengthened, forming cell assemblies and phase sequences. The former, on a shorter scale, are ensembles of synchronized cells that function transiently as a closed processing system; the latter, on a larger scale, correspond to the sequential activation of cell assemblies able to represent percepts and behaviors. Nowadays, the recording of large neuronal populations allows for the detection of multiple cell assemblies. Within Hebb's theory, the next logical step is the analysis of phase sequences. Here we detected phase sequences as consecutive assembly activation patterns, and then analyzed their graph attributes in relation to behavior. We investigated action potentials recorded from the adult rat hippocampus and neocortex before, during and after novel object exploration (experimental periods). Within assembly graphs, each assembly corresponded to a node, and each edge corresponded to the temporal sequence of consecutive node activations. The sum of all assembly activations was proportional to firing rates, but the activity of individual assemblies was not. Assembly repertoire was stable across experimental periods, suggesting that novel experience does not create new assemblies in the adult rat. Assembly graph attributes, on the other hand, varied significantly across behavioral states and experimental periods, and were separable enough to correctly classify experimental periods (Naïve Bayes classifier; maximum AUROCs ranging from 0.55 to 0.99) and behavioral states (waking, slow wave sleep, and rapid eye movement sleep; maximum AUROCs ranging from 0.64 to 0.98). Our findings agree with Hebb's view that assemblies correspond to primitive building blocks of representation, nearly unchanged in the adult, while phase sequences are labile across behavioral states and change after novel experience. The results are compatible with a role for phase sequences in behavior and cognition.

Signal-flow graphs: Direct method of reduction and MATLAB implementation

Block diagrams and signal-flow graphs are used to represent and to obtain the transfer function of interconnected systems. The reduction of signal-flow graphs is considered simpler than the reduction of block diagrams for systems with complex interrelationships. Signal-flow graphs reduction can be made without graphic manipulations of diagrams, and it is attractive for a computational implementation. In this paper the authors propose a computational method for direct reduction of signal-flow graphs. This method uses results presented in this paper about the calculation of literal determinants without symbolic mathematics tools. The Cramer's rule is applied for the solution of a set of linear equations, A program in MATLAB language for reduction of signal-flow graphs with the proposed method is presented.

Variance dispersion graphs for comparing blocked response surface designs

In this paper, we extend the use of the variance dispersion graph (VDG) to experiments in which the response surface (RS) design must be blocked. Through several examples we evaluate the prediction performances of RS designs in non-orthogonal block designs compared with the equivalent unblocked designs and orthogonally blocked designs. These examples illustrate that good prediction performance of designs in small blocks can be expected in practice. Most importantly, we show that the allocation of the treatment set to blocks can seriously affect the prediction properties of designs; thus, much care is needed in performing this allocation.