14,109 research outputs found

    Projected Power Iteration for Network Alignment

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    The network alignment problem asks for the best correspondence between two given graphs, so that the largest possible number of edges are matched. This problem appears in many scientific problems (like the study of protein-protein interactions) and it is very closely related to the quadratic assignment problem which has graph isomorphism, traveling salesman and minimum bisection problems as particular cases. The graph matching problem is NP-hard in general. However, under some restrictive models for the graphs, algorithms can approximate the alignment efficiently. In that spirit the recent work by Feizi and collaborators introduce EigenAlign, a fast spectral method with convergence guarantees for Erd\H{o}s-Reny\'i graphs. In this work we propose the algorithm Projected Power Alignment, which is a projected power iteration version of EigenAlign. We numerically show it improves the recovery rates of EigenAlign and we describe the theory that may be used to provide performance guarantees for Projected Power Alignment.Comment: 8 page

    Additive Approximation Algorithms for Modularity Maximization

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    The modularity is a quality function in community detection, which was introduced by Newman and Girvan (2004). Community detection in graphs is now often conducted through modularity maximization: given an undirected graph G=(V,E)G=(V,E), we are asked to find a partition C\mathcal{C} of VV that maximizes the modularity. Although numerous algorithms have been developed to date, most of them have no theoretical approximation guarantee. Recently, to overcome this issue, the design of modularity maximization algorithms with provable approximation guarantees has attracted significant attention in the computer science community. In this study, we further investigate the approximability of modularity maximization. More specifically, we propose a polynomial-time (cos(354π)1+58)\left(\cos\left(\frac{3-\sqrt{5}}{4}\pi\right) - \frac{1+\sqrt{5}}{8}\right)-additive approximation algorithm for the modularity maximization problem. Note here that cos(354π)1+58<0.42084\cos\left(\frac{3-\sqrt{5}}{4}\pi\right) - \frac{1+\sqrt{5}}{8} < 0.42084 holds. This improves the current best additive approximation error of 0.46720.4672, which was recently provided by Dinh, Li, and Thai (2015). Interestingly, our analysis also demonstrates that the proposed algorithm obtains a nearly-optimal solution for any instance with a very high modularity value. Moreover, we propose a polynomial-time 0.165980.16598-additive approximation algorithm for the maximum modularity cut problem. It should be noted that this is the first non-trivial approximability result for the problem. Finally, we demonstrate that our approximation algorithm can be extended to some related problems.Comment: 23 pages, 4 figure

    Improving Efficiency and Scalability of Sum of Squares Optimization: Recent Advances and Limitations

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    It is well-known that any sum of squares (SOS) program can be cast as a semidefinite program (SDP) of a particular structure and that therein lies the computational bottleneck for SOS programs, as the SDPs generated by this procedure are large and costly to solve when the polynomials involved in the SOS programs have a large number of variables and degree. In this paper, we review SOS optimization techniques and present two new methods for improving their computational efficiency. The first method leverages the sparsity of the underlying SDP to obtain computational speed-ups. Further improvements can be obtained if the coefficients of the polynomials that describe the problem have a particular sparsity pattern, called chordal sparsity. The second method bypasses semidefinite programming altogether and relies instead on solving a sequence of more tractable convex programs, namely linear and second order cone programs. This opens up the question as to how well one can approximate the cone of SOS polynomials by second order representable cones. In the last part of the paper, we present some recent negative results related to this question.Comment: Tutorial for CDC 201

    Scalable Approach to Uncertainty Quantification and Robust Design of Interconnected Dynamical Systems

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    Development of robust dynamical systems and networks such as autonomous aircraft systems capable of accomplishing complex missions faces challenges due to the dynamically evolving uncertainties coming from model uncertainties, necessity to operate in a hostile cluttered urban environment, and the distributed and dynamic nature of the communication and computation resources. Model-based robust design is difficult because of the complexity of the hybrid dynamic models including continuous vehicle dynamics, the discrete models of computations and communications, and the size of the problem. We will overview recent advances in methodology and tools to model, analyze, and design robust autonomous aerospace systems operating in uncertain environment, with stress on efficient uncertainty quantification and robust design using the case studies of the mission including model-based target tracking and search, and trajectory planning in uncertain urban environment. To show that the methodology is generally applicable to uncertain dynamical systems, we will also show examples of application of the new methods to efficient uncertainty quantification of energy usage in buildings, and stability assessment of interconnected power networks

    Toric algebra of hypergraphs

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    The edges of any hypergraph parametrize a monomial algebra called the edge subring of the hypergraph. We study presentation ideals of these edge subrings, and describe their generators in terms of balanced walks on hypergraphs. Our results generalize those for the defining ideals of edge subrings of graphs, which are well-known in the commutative algebra community, and popular in the algebraic statistics community. One of the motivations for studying toric ideals of hypergraphs comes from algebraic statistics, where generators of the toric ideal give a basis for random walks on fibers of the statistical model specified by the hypergraph. Further, understanding the structure of the generators gives insight into the model geometry.Comment: Section 3 is new: it explains connections to log-linear models in algebraic statistics and to combinatorial discrepancy. Section 6 (open problems) has been moderately revise
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