In this paper, we give a (polynomial-time) 3-approximation algorithm for the rooted subtree prune and regraft distance between two phylogenetic trees. This problem is known to be NP-complete and the best previously known approximation algorithm is a 5-approximation. We also give a faster fixed-parameter algorithm for the rooted subtree prune and regraft distance than was previously known.

We present a general approach to obtain direct and inverse results for approximation in Hölder norms. This approach is used to obtain a collection of new results related with estimates of the best polynomial approximation and with the approximation by linear operators of non-periodic functions in Hölder norms. © 2005 Elsevier Inc. All rights reserved.

We address the classical knapsack problem and a variant in which an upper bound is imposed on the number of items that can be selected. We show that appropriate combinations of rounding techniques yield novel and powerful ways of rounding. As an application of these techniques, we present a linear-storage Polynomial Time Approximation Scheme (PTAS) and a Fully Polynomial Time Approximation Sche...

In this article, Ritz approximation have been employed to obtain the numerical solutions of a class of the fractional optimal control problems based on the Caputo fractional derivative. Using polynomial basis functions, we obtain a system of nonlinear algebraic equations. This nonlinear system of equation is solved and the coefficients of basis polynomial are derived. The convergence of the num...

I first met Wolfgang Dahmen in 1974 in Oberwolfach. He looked like a high school student to me but he impressed everyone with his talk on whether polynomial operators could produce both polynomial and spectral orders of approximation. We became the best of friends and frequent collaborators. While Wolfgang’s mathematical contributions spread across many disciplines, a major thread in his work h...

A general method for near-best approximations to functionals on Rd, using scattered-data information is discussed. The method is actually the moving least-squares method, presented by the Backus-Gilbert approach. It is shown that the method works very well for interpolation, smoothing and derivatives’ approximations. For the interpolation problem this approach gives Mclain’s method. The method ...

We design a polynomial time 8/7-approximation algorithm for the Traveling Salesman Problem in which all distances are either one or two. This improves over the best known approximation factor for that problem. As a direct application we get a 7/6-approximation algorithm for the Maximum Path Cover Problem, similarly improving upon the best known approximation factor for that problem. The result ...

Nonlinear piecewise polynomial approximation: Theory and Algorithms Borislav Karaivanov We study nonlinear n-term approximation in Lp(R) (0 < p < ∞) from Courant elements or (discontinuous) piecewise polynomials generated by multilevel nested triangulations of R which allow arbitrarily sharp angles. To characterize the rate of approximation we introduce and develop three families of smoothness ...