5/4-Approximation of Minimum 2-Edge-Connected Spanning Subgraph

We provide a $5/4$-approximation algorithm for the minimum 2-edge-connected spanning subgraph problem. This improves upon the previous best ratio of $4/3$. The algorithm is based on applying local improvement steps on a starting solution provided by a standard ear decomposition together with the idea of running several iterations on residual graphs by excluding certain edges that do not belong to an optimum solution. The latter idea is a novel one, which allows us to bypass $3$-ears with no loss in approximation ratio, the bottleneck for obtaining a performance guarantee below $3/2$. Our algorithm also implies a simpler $7/4$-approximation algorithm for the matching augmentation problem, which was recently treated.Comment: The modification of 5-ears, which was both erroneous and unnecessary, is omitte

Scheme-theoretic Approach to Computational Complexity I. The Separation of P and NP

We lay the foundations of a new theory for algorithms and computational complexity by parameterizing the instances of a computational problem as a moduli scheme. Considering the geometry of the scheme associated to 3-SAT, we separate P and NP.Comment: 11 pages, corrections upon the referee repor

4/3-Approximation of Graphic TSP

We describe a $\frac{4}{3}$-approximation algorithm for the traveling salesman problem in which the distances between points are induced by graph-theoretical distances in an unweighted graph. The algorithm is based on finding a minimum cost perfect matching on the odd degree vertices of a carefully computed 2-edge-connected spanning subgraph.Comment: 10 pages, decomposition specified more carefully, Lemma 3 (now Lemma 2) correcte

Dual Growth with Variable Rates: An Improved Integrality Gap for Steiner Tree

A promising approach for obtaining improved approximation algorithms for Steiner tree is to use the bidirected cut relaxation (BCR). The integrality gap of this relaxation is at least $36/31$, and it has long been conjectured that its true value is very close to this lower bound. However, the best upper bound for general graphs is still $2$. With the aim of circumventing the asymmetric nature of BCR, Chakrabarty, Devanur and Vazirani [Math. Program., 130 (2011), pp. 1--32] introduced the simplex-embedding LP, which is equivalent to it. Using this, they gave a $\sqrt{2}$-approximation algorithm for quasi-bipartite graphs and showed that the integrality gap of the relaxation is at most $4/3$ for this class of graphs. In this paper, we extend the approach provided by these authors and show that the integrality gap of BCR is at most $7/6$ on quasi-bipartite graphs via a fast combinatorial algorithm. In doing so, we introduce a general technique, in particular a potentially widely applicable extension of the primal-dual schema. Roughly speaking, we apply the schema twice with variable rates of growth for the duals in the second phase, where the rates depend on the degrees of the duals computed in the first phase. This technique breaks the disadvantage of increasing dual variables in a monotone manner and creates a larger total dual value, thus presumably attaining the true integrality gap.Comment: A completely rewritten version of a previously retracted manuscript, using the simplex-embedding LP. The idea of growing duals with variable rates is still there. 23 pages, 7 figure

Scheme-theoretic Approach to Computational Complexity II. The Separation of P and NP over C\mathbb{C}, R\mathbb{R}, and Z\mathbb{Z}

We show that the problem of determining the feasibility of quadratic systems over $\mathbb{C}$, $\mathbb{R}$, and $\mathbb{Z}$ requires exponential time. This separates P and NP over these fields/rings in the BCSS model of computation.Comment: 4 pages. arXiv admin note: text overlap with arXiv:2107.0738

Estimation of hourly mean ambient temperatures with artificial neural networks

In this study, the artificial neural networks have been used for the estimation of hourly ambient temperature in Denizli, Turkey. The model was trained and tested with four years (2002-2005) of hourly mean temperature values. The hourly temperature values for the years 2002-2004 were used in training phase, the values for the year 2005 were used to test the model. The architecture of the ANN model was the multi-layer feedforward architecture and has three layers. Inputs of the network were month, day, hour, and two hourly mean temperatures at the previous hours, and the output was the mean temperature at the hour specified in the input. In the model, Levenberg-Marquardt learning algorithm which is a variant of backpropagation was used. With the software developed in Matlab, an ANN was constructed, trained, and tested for a different number of neurons in its hidden layer. The best result was obtained for 27 neurons, where R2, RMSE and MAPE values were found to be 0.99999, 0.92024 and 0.20900% for training, and 0.9999, 0.91301 and 0.20907% for test. The results show that the artificial neural network is powerful an alternate method in temperature estimations. © Association for Scientific Research

Quadrature Strategies for Constructing Polynomial Approximations

Finding suitable points for multivariate polynomial interpolation and approximation is a challenging task. Yet, despite this challenge, there has been tremendous research dedicated to this singular cause. In this paper, we begin by reviewing classical methods for finding suitable quadrature points for polynomial approximation in both the univariate and multivariate setting. Then, we categorize recent advances into those that propose a new sampling approach and those centered on an optimization strategy. The sampling approaches yield a favorable discretization of the domain, while the optimization methods pick a subset of the discretized samples that minimize certain objectives. While not all strategies follow this two-stage approach, most do. Sampling techniques covered include subsampling quadratures, Christoffel, induced and Monte Carlo methods. Optimization methods discussed range from linear programming ideas and Newton's method to greedy procedures from numerical linear algebra. Our exposition is aided by examples that implement some of the aforementioned strategies

Molecular mechanism of dynein recruitment to kinetochores by the Rod-Zw10-Zwilch complex and Spindly

The molecular motor dynein concentrates at the kinetochore region of mitotic chromosomes in animals to accelerate spindle microtubule capture and to control spindle checkpoint signaling. In this study, we describe the molecular mechanism used by the Rod-Zw10-Zwilch complex and the adaptor Spindly to recruit dynein to kinetochores in Caenorhabditis elegans embryos and human cells. We show that Rod's N-terminal beta-propeller and the associated Zwilch subunit bind Spindly's C-terminal domain, and we identify a specific Zwilch mutant that abrogates Spindly and dynein recruitment in vivo and Spindly binding to a Rod beta-propeller-Zwilch complex in vitro. Spindly's N-terminal coiled-coil uses distinct motifs to bind dynein light intermediate chain and the pointed-end complex of dynactin. Mutations in these motifs inhibit assembly of a dynein-dynactin-Spindly complex, and a null mutant of the dynactin pointed-end subunit p27 prevents kinetochore recruitment of dynein-dynactin without affecting other mitotic functions of the motor. Conservation of Spindly-like motifs in adaptors involved in intracellular transport suggests a common mechanism for linking dynein to cargo.This work was supported by a European Research Council Starting Grant (Dyneinome 338410) and a European Molecular Biology Organization Installation Grant to R. Gassmann. This work was also supported by funding from the Fundacao para a Ciencia e a Tecnologia to R. Gassmann (IF/01015/2013/CP1157/CT0006), C. Pereira (SFRH_BPD_95648_2013), and D.J. Barbosa (SFRH_BPD_101898_2014). Some C. elegans strains were provided by the Caenorhabditis Genetics Center, which is funded by the National Institutes of Health Office of Research Infrastructure Programs (P40 OD010440)