Given a hypergraph @math , the Minimum Connectivity Inference problem asks for a graph on the same vertex set as @math with the minimum number of edges such that the subgraph induced by every hyperedge of @math is connected. This problem has received a lot of attention these recent years, both from a theoretical and practical perspective, leading to several implemented approximation, greedy and heuristic algorithms. Concerning exact algorithms, only Mixed Integer Linear Programming (MILP) formulations have been experimented, all representing connectivity constraints by the means of graph flows. In this work, we investigate the efficiency of a constraint generation algorithm, where we iteratively add cut constraints to a simple ILP until a feasible (and optimal) solution is found. It turns out that our method is faster than the previous best flow-based MILP algorithm on random generated instances, which suggests that a constraint generation approach might be also useful for other optimization problems dealing with connectivity constraints. At last, we present the results of an enumeration algorithm for the problem.
Concerning the implementation of algorithms, previous works mainly focused on approximation, greedy and other heuristic techniques @cite_4 . To the best of our knowledge, the first exact algorithm was designed by Agarwal al @cite_17 @cite_7 in the context of structural biology, where the sought graph represents the contact relations between proteins of a macro-molecule, which has to be inferred from a hypergraph constructed by chemical experiments and mass spectrometry. In this work, the authors define a Mixed Integer Linear Programming (MILP) formulation of the problem, representing the connectivity constraints by flows. They also provide an enumeration method using their algorithm as a black box, by iteratively adding constraints to the MILP in order to forbid already found solutions. Both their optimization and enumeration algorithms were tested on some real-life (from a structural biology perspective) instances for which the contact graph was already known.
@cite_4 @cite_7 @cite_17
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Motivation: With the exponential growth of expression and protein–protein interaction (PPI) data, the frontier of research in systems biology shifts more and more to the integrated analysis of these large datasets. Of particular interest is the identification of functional modules in PPI networks, sharing common cellular function beyond the scope of classical pathways, by means of detecting differentially expressed regions in PPI networks. This requires on the one hand an adequate scoring of the nodes in the network to be identified and on the other hand the availability of an effective algorithm to find the maximally scoring network regions. Various heuristic approaches have been proposed in the literature. Results: Here we present the first exact solution for this problem, which is based on integer-linear programming and its connection to the well-known prize-collecting Steiner tree problem from Operations Research. Despite the NP-hardness of the underlying combinatorial problem, our method typically computes provably optimal subnetworks in large PPI networks in a few minutes. An essential ingredient of our approach is a scoring function defined on network nodes. We propose a new additive score with two desirable properties: (i) it is scalable by a statistically interpretable parameter and (ii) it allows a smooth integration of data from various sources. We apply our method to a well-established lymphoma microarray dataset in combination with associated survival data and the large interaction network of HPRD to identify functional modules by computing optimal-scoring subnetworks. In particular, we find a functional interaction module associated with proliferation over-expressed in the aggressive ABC subtype as well as modules derived from non-malignant by-stander cells. Availability: Our software is available freely for non-commercial purposes at http: www.planet-lisa.net. Contact: tobias.mueller@biozentrum.uni-wuerzburg.de Motivation: Inferring networks of proteins from biological data is a central issue of computational biology. Most network inference methods, including Bayesian networks, take unsupervised approaches in which the network is totally unknown in the beginning, and all the edges have to be predicted. A more realistic supervised framework, proposed recently, assumes that a substantial part of the network is known. We propose a new kernel-based method for supervised graph inference based on multiple types of biological datasets such as gene expression, phylogenetic profiles and amino acid sequences. Notably, our method assigns a weight to each type of dataset and thereby selects informative ones. Data selection is useful for reducing data collection costs. For example, when a similar network inference problem must be solved for other organisms, the dataset excluded by our algorithm need not be collected. Results: First, we formulate supervised network inference as a kernel matrix completion problem, where the inference of edges boils down to estimation of missing entries of a kernel matrix. Then, an expectation--maximization algorithm is proposed to simultaneously infer the missing entries of the kernel matrix and the weights of multiple datasets. By introducing the weights, we can integrate multiple datasets selectively and thereby exclude irrelevant and noisy datasets. Our approach is favorably tested in two biological networks: a metabolic network and a protein interaction network. Availability: Software is available on request. Contact: kato-tsuyoshi@aist.go.jp Supplementary information: A supplementary report including mathematical details is available at www.cbrc.jp kato faem faem.html We study the quantitative geometry of graphs in terms of their genus, using the structure of certain "cut graphs," i.e. subgraphs whose removal leaves a planar graph. In particular, we give optimal bounds for random partitioning schemes, as well as various types of embeddings. Using these geometric primitives, we present exponentially improved dependence on genus for a number of problems like approximate max-flow min-cut theorems, approximations for uniform and nonuniform Sparsest Cut, treewidth approximation, Laplacian eigenvalue bounds, and Lipschitz extension theorems and related metric labeling problems. We list here a sample of these improvements. All the following statements refer to graphs of genus g, unless otherwise noted. • We show that such graphs admit an O(log g)-approximate multi-commodity max-flow min-cut theorem for the case of uniform demands. This bound is optimal, and improves over the previous bound of O(g) [KPR93, FT03]. For general demands, we show that the worst possible gap is O(log g + CP), where CP is the gap for planar graphs. This dependence is optimal, and already yields a bound of O(log g + √log n), improving over the previous bound of O(√g log n) [KLMN04]. • We give an O(√log g)-approximation for the uniform Sparsest Cut, balanced vertex separator, and treewidth problems, improving over the previous bound of O(g) [FHL05]. • If a graph G has genus g and maximum degree D, we show that the kth Laplacian eigenvalue of G is (log g)2 · O(kgD n), improving over the previous bound of g2·O(kgD n) [KLPT09]. There is a lower bound of Ω(kgD n), making this result almost tight. • We show that if (X, d) is the shortest-path metric on a graph of genus g and S ⊆ X, then every L-Lipschitz map f: S → Z into a Banach space Z admits an O(L log g)-Lipschitz extension f: X → Z. This improves over the previous bound of O(Lg) [LN05], and compares to a lower bound of Ω(L√log g). In a related way, we show that there is an O(log g)-approximation for the 0-extension problem on such graphs, improving over the previous O(g) bound. • We show that every n-vertex shortest-path metric on a graph of genus g embeds into L2 with distortion O(log g + √log n), improving over the previous bound of O(√g log n). Our result is asymptotically optimal for every dependence g = g(n). The greedy sequential algorithm for maximal independent set (MIS) loops over the vertices in an arbitrary order adding a vertex to the resulting set if and only if no previous neighboring vertex has been added. In this loop, as in many sequential loops, each iterate will only depend on a subset of the previous iterates (i.e. knowing that any one of a vertex's previous neighbors is in the MIS, or knowing that it has no previous neighbors, is sufficient to decide its fate one way or the other). This leads to a dependence structure among the iterates. If this structure is shallow then running the iterates in parallel while respecting the dependencies can lead to an efficient parallel implementation mimicking the sequential algorithm. In this paper, we show that for any graph, and for a random ordering of the vertices, the dependence length of the sequential greedy MIS algorithm is polylogarithmic (O(log^2 n) with high probability). Our results extend previous results that show polylogarithmic bounds only for random graphs. We show similar results for greedy maximal matching (MM). For both problems we describe simple linear-work parallel algorithms based on the approach. The algorithms allow for a smooth tradeoff between more parallelism and reduced work, but always return the same result as the sequential greedy algorithms. We present experimental results that demonstrate efficiency and the tradeoff between work and parallelism.