We give a polynomial time approximation scheme (PTAS) for the recti-linear Steiner arborescence problem in the plane. The result is obtained by modifying Arora's PTAS for Euclidean TSP. The previously best known result was a 2-approximation algorithm.

For five single machine total tardiness problems a fully polynomial-time approximation scheme (FPTAS) based on a graphical algorithm is presented. The FPTAS has the best running time among the known approximation schemes for these problems.

Recall that the approximation ratio for an approximation algorithm is a measure to evaluate the approximation performance of the algorithm. The closer the ratio to 1 the better the approximation performance of the algorithm. It is notable that there is a class of NP-hard optimization problems, most originating from scheduling problems, for which there are polynomial time approximation algorithm...

we obtain a sucint and nesessery theoreoms simple for compactness andweakly compactness of the best approximate sets by closed unit balls. also weconsider relations kadec-klee property and shur property with this objects.these theorems are extend of papers mohebi and narayana.

We present a new method that allows us to get a direct proof of the classical Bernstein asymptotics for the error of the best uniform polynomial approximation of |x| on two symmetric intervals. Note, that in addition, we get asymptotics for the polynomials themselves under a certain renormalization. Also, we solve a problem on asymptotics of the best approximation of sgn (x) on [−1,−a] ∪ [a, 1]...