Choose Outsiders First: a mean 2-approximation random algorithm for covering problems

A high number of discrete optimization problems, including Vertex Cover, Set Cover or Feedback Vertex Set, can be unified into the class of covering problems. Several of them were shown to be inapproximable by deterministic algorithms. This article proposes a new random approach, called Choose Outsiders First, which consists in selecting randomly ele- ments which are excluded from the cover. We show that this approach leads to random outputs which mean size is at most twice the optimal solution.Comment: 8 pages The paper has been withdrawn due to an error in the proo

Overlapping stochastic block models with application to the French political blogosphere

Complex systems in nature and in society are often represented as networks, describing the rich set of interactions between objects of interest. Many deterministic and probabilistic clustering methods have been developed to analyze such structures. Given a network, almost all of them partition the vertices into disjoint clusters, according to their connection profile. However, recent studies have shown that these techniques were too restrictive and that most of the existing networks contained overlapping clusters. To tackle this issue, we present in this paper the Overlapping Stochastic Block Model. Our approach allows the vertices to belong to multiple clusters, and, to some extent, generalizes the well-known Stochastic Block Model [Nowicki and Snijders (2001)]. We show that the model is generically identifiable within classes of equivalence and we propose an approximate inference procedure, based on global and local variational techniques. Using toy data sets as well as the French Political Blogosphere network and the transcriptional network of Saccharomyces cerevisiae, we compare our work with other approaches.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS382 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Incremental complexity of a bi-objective hypergraph transversal problem

The hypergraph transversal problem has been intensively studied, from both a theoretical and a practical point of view. In particular , its incremental complexity is known to be quasi-polynomial in general and polynomial for bounded hypergraphs. Recent applications in computational biology however require to solve a generalization of this problem, that we call bi-objective transversal problem. The instance is in this case composed of a pair of hypergraphs (A, B), and the aim is to find minimal sets which hit all the hyperedges of A while intersecting a minimal set of hyperedges of B. In this paper, we formalize this problem, link it to a problem on monotone boolean $\land$ -- $\lor$ formulae of depth 3 and study its incremental complexity

Minimum Eccentricity Shortest Path Problem: an Approximation Algorithm and Relation with the k-Laminarity Problem

The Minimum Eccentricity Shortest Path (MESP) Problem consists in determining a shortest path (a path whose length is the distance between its extremities) of minimum eccentricity in a graph. It was introduced by Dragan and Leitert [9] who described a linear-time algorithm which is an 8-approximation of the problem. In this paper, we study deeper the double-BFS procedure used in that algorithm and extend it to obtain a linear-time 3-approximation algorithm. We moreover study the link between the MESP problem and the notion of laminarity, introduced by Völkel et al [12], corresponding to its restriction to a diameter (i.e. a shortest path of maximum length), and show tight bounds between MESP and laminarity parameters

A model for gene deregulation detection using expression data

In tumoral cells, gene regulation mechanisms are severely altered, and these modifications in the regulations may be characteristic of different subtypes of cancer. However, these alterations do not necessarily induce differential expressions between the subtypes. To answer this question, we propose a statistical methodology to identify the misregulated genes given a reference network and gene expression data. Our model is based on a regulatory process in which all genes are allowed to be deregulated. We derive an EM algorithm where the hidden variables correspond to the status (under/over/normally expressed) of the genes and where the E-step is solved thanks to a message passing algorithm. Our procedure provides posterior probabilities of deregulation in a given sample for each gene. We assess the performance of our method by numerical experiments on simulations and on a bladder cancer data set

Arc-chromatic number of digraphs in which each vertex has bounded outdegree or bounded indegree

A $k$-digraph is a digraph in which every vertex has outdegree at most $k$. A (k\veel)-digraph is a digraph in which a vertex has either outdegree at most $k$ or indegree at most $l$. Motivated by function theory, we study the maximum value $\Phi(k)$ (resp. $\Phi^\vee(k,l)$ of the arc-chromatic number over the $k$-digraphs (resp. (k\veel)-digraphs). El-Sahili showed that $\Phi^{\vee}(k,k)\leq 2k+1$. After giving a simple proof of this result, we show some better bounds. We show $\max\{\log(2k+3), \theta(k+1)\}\leq \Phi(k)\leq \theta(2k)$ and $\max\{\log(2k+2l+4), \theta(k+1), \theta(l+1)\}\leq \Phi^{\vee}(k,l)\leq \theta(2k+2l)$ where $\theta$ is the function defined by $\theta(k)=\min\{s : {s\choose \left\lceil \frac{s}{2}\right\rceil}\geq k\}$. We then study in more details properties of $\Phi$ and $\Phi^{\vee}$. Finally, we give the exact values of $\Phi(k)$ and $\Phi^{\vee}(k,l)$ for $l\leq k\leq 3$

Enumerating Chemical Organisations in Consistent Metabolic Networks: Complexity and Algorithms

International audienceThe structural analysis of metabolic networks aims both at understanding the function and the evolution of metabolism. While it is commonly admitted that metabolism is modular, the identi cation of metabolic modules remains an open topic. Several de nitions of what is a module have been proposed. We focus here on the notion of chemical organisations, i.e. sets of molecules which are closed and self-maintaining. We show that nding a reactive organisation is NP-hard even if the network is mass- and ux-consistent and that the hardness comes from blocking cycles. We then propose new algorithms for enumerating chemical organisations that are theoretically more e cient than existing approaches

Efficient Enumeration of Bipartite Subgraphs in Graphs

Subgraph enumeration problems ask to output all subgraphs of an input graph that belongs to the specified graph class or satisfy the given constraint. These problems have been widely studied in theoretical computer science. As far, many efficient enumeration algorithms for the fundamental substructures such as spanning trees, cycles, and paths, have been developed. This paper addresses the enumeration problem of bipartite subgraphs. Even though bipartite graphs are quite fundamental and have numerous applications in both theory and application, its enumeration algorithms have not been intensively studied, to the best of our knowledge. We propose the first non-trivial algorithms for enumerating all bipartite subgraphs in a given graph. As the main results, we develop two efficient algorithms: the one enumerates all bipartite induced subgraphs of a graph with degeneracy $k$ in $O(k)$ time per solution. The other enumerates all bipartite subgraphs in $O(1)$ time per solution

Overlapping clustering methods for networks

Networks allow the representation of interactions between objects. Their structures are often complex to explore and need some algorithmic and statistical tools for summarizing. One possible way to go is to cluster their vertices into groups having similar connectivity patterns. This chapter aims at presenting an overview of clustering methods for network vertices. Common community structure searching algorithms are detailed. The well-known Stochastic Block Model (SBM) is then introduced and its generalization to overlapping mixed membership structure closes the chapter. Examples of application are also presented and the main hypothesis underlying the presented algorithms discussed