A general method for common intervals

Given an elementary chain of vertex set V, seen as a labelling of V by the set {1, ...,n=|V|}, and another discrete structure over $V$, say a graph G, the problem of common intervals is to compute the induced subgraphs G[I], such that $I$ is an interval of [1, n] and G[I] satisfies some property Pi (as for example Pi= "being connected"). This kind of problems comes from comparative genomic in bioinformatics, mainly when the graph $G$ is a chain or a tree (Heber and Stoye 2001, Heber and Savage 2005, Bergeron et al 2008). When the family of intervals is closed under intersection, we present here the combination of two approaches, namely the idea of potential beginning developed in Uno, Yagiura 2000 and Bui-Xuan et al 2005 and the notion of generator as defined in Bergeron et al 2008. This yields a very simple generic algorithm to compute all common intervals, which gives optimal algorithms in various applications. For example in the case where $G$ is a tree, our framework yields the first linear time algorithms for the two properties: "being connected" and "being a path". In the case where $G$ is a chain, the problem is known as: common intervals of two permutations (Uno and Yagiura 2000), our algorithm provides not only the set of all common intervals but also with some easy modifications a tree structure that represents this set

Assessing the Computational Complexity of Multi-Layer Subgraph Detection

Multi-layer graphs consist of several graphs (layers) over the same vertex set. They are motivated by real-world problems where entities (vertices) are associated via multiple types of relationships (edges in different layers). We chart the border of computational (in)tractability for the class of subgraph detection problems on multi-layer graphs, including fundamental problems such as maximum matching, finding certain clique relaxations (motivated by community detection), or path problems. Mostly encountering hardness results, sometimes even for two or three layers, we can also spot some islands of tractability

Unique perfect phylogeny is N P -hard

Abstract. We answer, in the affirmative, the following question proposed by Mike Steel as a $100 challenge: "Is the following problem N Phard? Given a ternary † phylogenetic X-tree T and a collection Q of quartet subtrees on X, is T the only tree that displays Q?" [28, 29] As a particular consequence of this, we show that the unique chordal sandwich problem is also N P -hard

Chemical analysis of polymer blends via synchrotron X-ray tomography

Material properties of industrial polymer blends are of great importance. X-ray tomography has been used to obtain spatial chemical information about various polymer blends. The spatial images are acquired with synchrotron X-ray tomography because of its rapidity, good spatial resolution, large field-of-view, and elemental sensitivity. The spatial absorption data acquired from X-ray tomography experiments is converted to spatial chemical information via a linear least squares fit of multi-spectral X-ray absorption data. A fiberglass-reinforced polymer blend with a new-generation flame retardant is studied with multi-energy synchrotron X-ray tomography to assess the blend homogeneity. Relative to other composite materials, this sample is difficult to image due to low x-ray contrast between the fiberglass reinforcement and the polymer blend. To investigate chemical composition surrounding the glass fibers, new procedures were developed to find and mark the fiberglass, then assess the flame retardant distribution near the fiber. Another polymer blending experiment using three-dimensional chemical analysis techniques to look at a polymer additive problem called blooming was done. To investigate the chemical process of blooming, new procedures are developed to assess the flame retardant distribution as a function of annealing time in the sample. With the spatial chemical distribution we fit the concentrations to a diffusion equation to each time step in the annealing process. Finally the diffusion properties of a polymer blend composed of hexabromobenzene and o-terphenyl was studied. The diffusion properties were compared with computer simulations of the blend

Competitive graph searches

AbstractWe exemplify an optimization criterion for divide-and-conquer algorithms with a technique called generic competitive graph search. The technique is then applied to solve two problems arising from biocomputing, so-called Common Connected Components and Cograph Sandwich. The first problem can be defined as follows: given two graphs on the same set of n vertices, find the coarsest partition of the vertex set into subsets which induce connected subgraphs in both input graphs. The second problem is an instance of sandwich problems: given a partial subgraph G1 of G2, find a partial subgraph G of G2 that is partial supergraph of G1 (sandwich), and that is a cograph. For the former problem our generic algorithm not only achieves the current best known performance on arbitrary graphs and forests, but also improves by a logn factor when the input is made of planar graphs. However, our complexity for intervals graphs is slightly lower than a recent result. For the latter problem, we first study the relationship between the common connected components problem and the cograph sandwich problem, then, using our competitive graph search paradigm, we improve the computation of cograph sandwiches from O(n(n+m)) down to O(n+mlog2n), where n is the number of vertices and m of total edges