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 the Erd\H{o}s-Tuza-Valtr Conjecture

The Erd\H{o}s-Szekeres conjecture states that any set of more than $2^{n-2}$ points in the plane with no three on a line contains the vertices of a convex $n$-gon. Erd\H{o}s, Tuza, and Valtr strengthened the conjecture by stating that any set of more than $\sum_{i = n - b}^{a - 2} \binom{n - 2}{i}$ points in a plane either contains the vertices of a convex $n$-gon, $a$ points lying on a concave downward curve, or $b$ points lying on a concave upward curve. They also showed that the generalization is actually equivalent to the Erd\H{o}s-Szekeres conjecture. We prove the first new case of the Erd\H{o}s-Tuza-Valtr conjecture since the original 1935 paper of Erd\H{o}s and Szekeres. Namely, we show that any set of $\binom{n-1}{2} + 2$ points in the plane with no three points on a line and no two points sharing the same $x$-coordinate either contains 4 points lying on a concave downward curve or the vertices of a convex $n$-gon.Comment: 16 pages, 8 figure

Contaje de triángulos en conjuntos de puntos coloreados: un problema de la geometría combinatoria

A classical object of study in combinatorial geometry are sets S of points in the plane. A triangle with vertices from S is called empty if it contains no points of S in its interior. The number of empty triangles depends on the positions of points from S and a burning question is: How many empty triangles are there at least, among all sets S of n points? In order to discard degenerate point configurations, we only consider sets S without three collinear points. In this project, a software has been developed which allows to count the number of empty triangles in a set of n points in the plane. The software permits generation of point sets and their graphical visualization, as well as searching and displaying of optimal point configurations encountered. A point set of a given cardinality is said to be optimal if it contains the minimum number of empty triangles. The objective is to derive bounds on the minimum number of empty triangles by means of experiments realized with our software. The created program also allows to count empty monochromatic triangles in two-colored point sets. A triangle is called monochromatic if its three vertices have the same color. While the first problem has been studied extensively during the last decades, the two-colored version remains to be explored in depth. In this work we also expose our results on the minimum number of empty triangles in (small) two-colored point sets. Also, the treated problem is put in context with related results, such as the Erdös-Szekeres theorem, and a short outline of famous problems which contributed to the rise of combinatorial geometry is presented.Un objeto clásico de estudio en la Geometría combinatoria son conjuntos S de n puntos en el plano. Se dice que un triángulo con vértices en S esta vacío si no contiene puntos de S en su interior. El número de triángulos vacíos depende de cómo se dibujó el conjunto S y una pregunta ardiente es: ¿Cuántos triángulos vacíos hay como mínimo en cada conjunto S de n puntos? Para descartar configuraciones de puntos degeneradas solo se consideran nubes de puntos sin tres puntos colineales

On the Minimum Size of a Point Set Containing a 5-Hole and a Disjoint 4-Hole

Let H(k; l), k ≤ l denote the smallest integer such that any set of H(k; l) points in the plane, no three on a line, contains an empty convex k-gon and an empty convex l-gon, which are disjoint, that is, their convex hulls do not intersect. Hosono and Urabe [JCDCG, LNCS 3742, 117–122, 2004] proved that 12 ≤ H(4, 5) ≤ 14. Very recently, using a Ramseytype result for disjoint empty convex polygons proved by Aichholzer et al. [Graphs and Combinatorics, Vol. 23, 481–507, 2007], Hosono and Urabe [Kyoto CGGT, LNCS 4535, 90–100, 2008] improve the upper bound to 13. In this paper, with the help of the same Ramsey-type result, we prove that H(4; 5) = 12

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