Bounds on the maximum multiplicity of some common geometric graphs

We obtain new lower and upper bounds for the maximum multiplicity of some weighted and, respectively, non-weighted common geometric graphs drawn on n points in the plane in general position (with no three points collinear): perfect matchings, spanning trees, spanning cycles (tours), and triangulations. (i) We present a new lower bound construction for the maximum number of triangulations a set of n points in general position can have. In particular, we show that a generalized double chain formed by two almost convex chains admits {\Omega}(8.65^n) different triangulations. This improves the bound {\Omega}(8.48^n) achieved by the double zig-zag chain configuration studied by Aichholzer et al. (ii) We present a new lower bound of {\Omega}(12.00^n) for the number of non-crossing spanning trees of the double chain composed of two convex chains. The previous bound, {\Omega}(10.42^n), stood unchanged for more than 10 years. (iii) Using a recent upper bound of 30^n for the number of triangulations, due to Sharir and Sheffer, we show that n points in the plane in general position admit at most O(68.62^n) non-crossing spanning cycles. (iv) We derive lower bounds for the number of maximum and minimum weighted geometric graphs (matchings, spanning trees, and tours). We show that the number of shortest non-crossing tours can be exponential in n. Likewise, we show that both the number of longest non-crossing tours and the number of longest non-crossing perfect matchings can be exponential in n. Moreover, we show that there are sets of n points in convex position with an exponential number of longest non-crossing spanning trees. For points in convex position we obtain tight bounds for the number of longest and shortest tours. We give a combinatorial characterization of the longest tours, which leads to an O(nlog n) time algorithm for computing them

Chaotic quasi-collision trajectories in the 3-centre problem

We study a particular kind of chaotic dynamics for the planar 3-centre problem on small negative energy level sets. We know that chaotic motions exist, if we make the assumption that one of the centres is far away from the other two (see Bolotin and Negrini, J. Diff. Eq. 190 (2003), 539--558): this result has been obtained by the use of the Poincar\'e-Melnikov theory. Here we change the assumption on the third centre: we do not make any hypothesis on its position, and we obtain a perturbation of the 2-centre problem by assuming its intensity to be very small. Then, for a dense subset of possible positions of the perturbing centre on the real plane, we prove the existence of uniformly hyperbolic invariant sets of periodic and chaotic almost collision orbits by the use of a general result of Bolotin and MacKay (see Cel. Mech. & Dyn. Astr. 77 (2000), 49--75). To apply it, we must preliminarily construct chains of collision arcs in a proper way. We succeed in doing that by the classical regularisation of the 2-centre problem and the use of the periodic orbits of the regularised problem passing through the third centre.Comment: 22 pages, 6 figure

Graph Connectivity in Noisy Sparse Subspace Clustering

Subspace clustering is the problem of clustering data points into a union of low-dimensional linear/affine subspaces. It is the mathematical abstraction of many important problems in computer vision, image processing and machine learning. A line of recent work (4, 19, 24, 20) provided strong theoretical guarantee for sparse subspace clustering (4), the state-of-the-art algorithm for subspace clustering, on both noiseless and noisy data sets. It was shown that under mild conditions, with high probability no two points from different subspaces are clustered together. Such guarantee, however, is not sufficient for the clustering to be correct, due to the notorious "graph connectivity problem" (15). In this paper, we investigate the graph connectivity problem for noisy sparse subspace clustering and show that a simple post-processing procedure is capable of delivering consistent clustering under certain "general position" or "restricted eigenvalue" assumptions. We also show that our condition is almost tight with adversarial noise perturbation by constructing a counter-example. These results provide the first exact clustering guarantee of noisy SSC for subspaces of dimension greater then 3.Comment: 14 pages. To appear in The 19th International Conference on Artificial Intelligence and Statistics, held at Cadiz, Spain in 201

Testing for Spanning with Futrures Contracts and Nontraded Assets: A General Approach

This paper generalizes the notion of mean-variance spanning as de- ned in the seminal paper of Huberman & Kandel (1987) in three di- mensions.It is shown how regression techniques can be used to test for spanning for more general classes of utility functions, in case some as- sets are nontraded, and in case some of the assets are zero-investment securities such as futures contracts.We then implement these tech- niques to test whether a basic set of three international stock indices, the S&P 500, the FAZ (Germany), and the FTSE (UK), span a set of commodity and currency futures contracts.Depending on whether mean-variance, logarithmic, or power utility functions are considered, the hypothesis of spanning can be rejected for most futures contracts considered.If an investor has a position in a nontraded commodity, then the hypothesis of spanning can almost always be rejected for fu- tures contracts on that commodity for all utility functions considered.For currency futures this is only the case for a power utility function that re ects a preference for skewness.Finally, if we explicitly take into account net futures positions of large traders that are known to have predictive power for futures returns, the hypothesis of spanning can be rejected for most futures contracts.regression analysis;futures

A Polyhedral Approximation Framework for Convex and Robust Distributed Optimization

In this paper we consider a general problem set-up for a wide class of convex and robust distributed optimization problems in peer-to-peer networks. In this set-up convex constraint sets are distributed to the network processors who have to compute the optimizer of a linear cost function subject to the constraints. We propose a novel fully distributed algorithm, named cutting-plane consensus, to solve the problem, based on an outer polyhedral approximation of the constraint sets. Processors running the algorithm compute and exchange linear approximations of their locally feasible sets. Independently of the number of processors in the network, each processor stores only a small number of linear constraints, making the algorithm scalable to large networks. The cutting-plane consensus algorithm is presented and analyzed for the general framework. Specifically, we prove that all processors running the algorithm agree on an optimizer of the global problem, and that the algorithm is tolerant to node and link failures as long as network connectivity is preserved. Then, the cutting plane consensus algorithm is specified to three different classes of distributed optimization problems, namely (i) inequality constrained problems, (ii) robust optimization problems, and (iii) almost separable optimization problems with separable objective functions and coupling constraints. For each one of these problem classes we solve a concrete problem that can be expressed in that framework and present computational results. That is, we show how to solve: position estimation in wireless sensor networks, a distributed robust linear program and, a distributed microgrid control problem.Comment: submitted to IEEE Transactions on Automatic Contro

Testing for Spanning with Futrures Contracts and Nontraded Assets:A General Approach

This paper generalizes the notion of mean-variance spanning as de- ned in the seminal paper of Huberman & Kandel (1987) in three di- mensions.It is shown how regression techniques can be used to test for spanning for more general classes of utility functions, in case some as- sets are nontraded, and in case some of the assets are zero-investment securities such as futures contracts.We then implement these tech- niques to test whether a basic set of three international stock indices, the S&P 500, the FAZ (Germany), and the FTSE (UK), span a set of commodity and currency futures contracts.Depending on whether mean-variance, logarithmic, or power utility functions are considered, the hypothesis of spanning can be rejected for most futures contracts considered.If an investor has a position in a nontraded commodity, then the hypothesis of spanning can almost always be rejected for fu- tures contracts on that commodity for all utility functions considered.For currency futures this is only the case for a power utility function that re ects a preference for skewness.Finally, if we explicitly take into account net futures positions of large traders that are known to have predictive power for futures returns, the hypothesis of spanning can be rejected for most futures contracts.