A linear time approximation algorithm for permutation flow shop scheduling

AbstractIn the last 40 years, the permutation flow shop scheduling (PFS) problem with makespan minimization has been a central problem, known for its intractability, that has been well studied from both theoretical and practical aspects. The currently best performance ratio of a deterministic approximation algorithm for the PFS was recently presented by Nagarajan and Sviridenko, using a connection between the PFS and the longest increasing subsequence problem. In a different and independent way, this paper employs monotone subsequences in the approximation analysis techniques. To do this, an extension of the Erdös–Szekeres theorem to weighted monotone subsequences is presented. The result is a simple deterministic algorithm for the PFS with a similar approximation guarantee, but a much lower time complexity

On Some Applications of Graph Theory, I

In a series of papers, of which the present one is Part 1, it is shown that solutions to a variety of problems in distance geometry, potential theory and theory of metric spaces are provided by appropriate applications of graph theoretic results. (c) 1972 Published by Elsevier B.V

On Consecutive Triples Of Powerful Numbers

A powerful number is a positive integer such that every prime that appears in its prime factorization appears there at least twice. Erdős, Mollin and Walsh conjectured that three consecutive powerful numbers do not exist. This paper shows that if they do exist, the smallest of the three numbers must have remainder 7, 27, or 35 when divided by 36

A Theorem of Barany Revisited and Extended

International audienceThe colorful Caratheodory theorem states that given d+1 sets of points in R^d, the convex hull of each containing the origin, there exists a simplex (called a 'rainbow simplex') with at most one point from each point set, which also contains the origin. Equivalently, either there is a hyperplane separating one of these d+1 sets of points from the origin, or there exists a rainbow simplex containing the origin. One of our results is the following extension of the colorful Caratheodory theorem: given d/2+1 sets of points in $R^d, and a convex object C, then either one set can be separated from C by a constant (depending only on d) number of hyperplanes, or there is a (d/2+1)-dimensional rainbow simplex intersecting C