Optimal designs for rational function regression

We consider optimal non-sequential designs for a large class of (linear and nonlinear) regression models involving polynomials and rational functions with heteroscedastic noise also given by a polynomial or rational weight function. The proposed method treats D-, E-, A-, and $\Phi_p$-optimal designs in a unified manner, and generates a polynomial whose zeros are the support points of the optimal approximate design, generalizing a number of previously known results of the same flavor. The method is based on a mathematical optimization model that can incorporate various criteria of optimality and can be solved efficiently by well established numerical optimization methods. In contrast to previous optimization-based methods proposed for similar design problems, it also has theoretical guarantee of its algorithmic efficiency; in fact, the running times of all numerical examples considered in the paper are negligible. The stability of the method is demonstrated in an example involving high degree polynomials. After discussing linear models, applications for finding locally optimal designs for nonlinear regression models involving rational functions are presented, then extensions to robust regression designs, and trigonometric regression are shown. As a corollary, an upper bound on the size of the support set of the minimally-supported optimal designs is also found. The method is of considerable practical importance, with the potential for instance to impact design software development. Further study of the optimality conditions of the main optimization model might also yield new theoretical insights.Comment: 25 pages. Previous version updated with more details in the theory and additional example

Improved Approximation Algorithms for Capacitated Facility Location Problems

In a recent surprising result, Korupolu, Plaxton, and Rajaraman [10, 11] showed that a simple local search heuristic for the capacitated facility location problem (CFLP) in which the service costs obey the triangle inequality produces a solution in polynomial time which is within a factor of 8 + ffl of the value of an optimal solution. By simplifying their analysis, we are able to show that the same heuristic produces solution which are within a factor of 6(1 + ffl) of the value of an optimal solution. Our simplified analysis uses the supermodularity of the cost function of the problem and the integrality of the transshipment polyhedron. Additionally, we consider the variant of the CFLP in which one may open multiple copies of any facility. Using ideas from the analysis of the local search heuristic, we show how to turn any ff-approximation algorithm for this variant into one which, at an additional cost of twice the optimal of the standard CFLP, opens at most one additional copy of an..

Characterizing the plasma metabolome during and following a maximal exercise cycling test

Although complex in nature, a number of metabolites have been implicated in the onset of exercise-induced fatigue. The purpose of this study was to identify changes in the plasma metabolome and specifically, to identify candidate metabolites associated with the onset of fatigue during prolonged cycling. Eighteen healthy and recreationally active men (mean ± SD; age: 24.7 ± 4.8 yr; mass 67.1 ± 6.1 kg; body mass index: 22.8 ± 2.2; peak oxygen uptake: 40.9 ± 6.1 ml·kg−1·min−1) were recruited to this study. Participants performed a prolonged cycling time-to-exhaustion (TTE) test at an intensity corresponding to a fixed blood lactate concentration (3 mmol/l). Plasma samples collected at 10 min of exercise, before fatigue (last sample before fatigue <10 min before fatigue), immediately after fatigue (point of exhaustion), and 20 min after fatigue were assessed using a liquid chromatography-mass spectrometry-based metabolomic approach. Eighty metabolites were putatively identified, with 68 metabolites demonstrating a significant change during the cycling task (duration: ~80.9 ± 13.6 min). A clear multivariate structure in the data was revealed, with the first principal component (36% total variance) describing a continuous increase in metabolite concentration throughout the TTE trial and recovery, whereas the second principal component (14% total variance) showed an increase in metabolite concentration followed by a recovery trajectory, peaking at the point of fatigue. Six clusters of correlated metabolites demonstrating unique metabolite trajectories were identified, including significant separation in the metabolome between prefatigue and postfatigue time points. In accordance with our hypothesis, free-fatty acids and tryptophan contributed to differences in the plasma metabolome at fatigue. NEW & NOTEWORTHY Metabolites have long been implicated in the onset of fatigue. This study applied a metabolomic approach to track 80 plasma-borne metabolites during a cycle to fatigue task. Of these, 68 metabolites demonstrated significant change, with the plasma metabolome at fatigue being clearly distinguishable from other time points. Six unique clusters of metabolites were identified, and free fatty acids were strongly associated with fatigue onset therein lending support to the central fatigue hypothesis

Characterizing the plasma metabolome during 14 days of live-high, train-low simulated altitude: A metabolomic approach

The purpose of this study was to determine the influence of 14 days of normobaric hypoxic simulated altitude exposure at 3000 m on the human plasma metabolomic profile. For 14 days, 10 well‐trained endurance runners (six men and four women; 29 ± 7 years of age) lived at 3000 m simulated altitude, accumulating 196.4 ± 25.6 h of hypoxic exposure, and trained at ∼600 m. Resting plasma samples were collected at baseline and on days 3 and 14 of altitude exposure and stored at −80°C. Plasma samples were analysed using liquid chromatography–high‐resolution mass spectrometry to construct a metabolite profile of altitude exposure. Mass spectrometry of plasma identified 36 metabolites, of which eight were statistically significant (false discovery rate probability 0.1) from baseline to either day 3 or day 14. Specifically, changes in plasma metabolites relating to amino acid metabolism (tyrosine and proline), glycolysis (adenosine) and purine metabolism (adenosine) were observed during altitude exposure. Principal component canonical variate analysis showed significant discrimination between group means (P < 0.05), with canonical variate 1 describing a non‐linear recovery trajectory from baseline to day 3 and then back to baseline by day 14. Conversely, canonical variate 2 described a weaker non‐recovery trajectory and increase from baseline to day 3, with a further increase from day 3 to 14. The present study demonstrates that metabolomics can be a useful tool to monitor metabolic changes associated with altitude exposure. Furthermore, it is apparent that altitude exposure alters multiple metabolic pathways, and the time course of these changes is different over 14 days of altitude exposure

Approximation Algorithms with Bounded Performance Guarantees for the Clustered Traveling Salesman Problem

Let G = (V, E) be a complete undirected graph with vertex set V , edge set E, and edge weights l(e) satisfying triangle inequality. The vertex set V is partitioned into clusters V 1 , ..., V k . The clustered traveling salesman problem (CTSP) is to compute a shortest Hamiltonian cycle (tour) that visits all the vertices, and in which the vertices of each cluster are visited consecutively. Since this problem is a generalization of the traveling salesman problem, it is NP-hard. In this paper, we consider several variants of this basic problem and provide polynomial time approximation algorithms for them

Vertex Cover in Graphs with Locally Few Colors

Abstract. In [13], Erdős et al. defined the local chromatic number of a graph as the minimum number of colors that must appear within distance 1 of a vertex. For any ∆ ≥ 2, there are graphs with arbitrarily large chromatic number that can be colored so that (i) no vertex neighborhood contains more than ∆ different colors (bounded local colorability), and (ii) adjacent vertices from two color classes induce a complete bipartite graph (biclique coloring). We investigate the weighted vertex cover problem in graphs when a lo-cally bounded coloring is given. This generalizes the vertex cover prob-lem in bounded degree graphs to a class of graphs with arbitrarily large chromatic number. Assuming the Unique Game Conjecture, we provide a tight characterization. We prove that it is UGC-hard to improve the approximation ratio of 2 − 2/(∆+ 1) if the given local coloring is not a biclique coloring. A matching upper bound is also provided. Vice versa, when properties (i) and (ii) hold, we present a randomized algorithm with approximation ratio of 2 − Ω(1) ln ln∆ ln∆. This matches known inap-proximability results for the special case of bounded degree graphs. Moreover, we show that the obtained result finds a natural application in a classical scheduling problem, namely the precedence constrained single machine scheduling problem to minimize the total weighted completion time. In a series of recent papers it was established that this scheduling problem is a special case of the minimum weighted vertex cover in graphs GP of incomparable pairs defined in the dimension theory of partial orders. We show that GP satisfies properties (i) and (ii) where ∆ − 1 is the maximum number of predecessors (or successors) of each job