In this paper, we study the asymmetric traveling salesman problem (ATSP) with strengthened triangle inequality, i.e. for some γ ∈ [ 1 2 , 1) the edge weights satisfy w(u, v) ≤ γ(w(u, x) + w(x, v)) for all distinct vertices u, v, x. We present two approximation algorithms for this problem. The first one is very simple and has approximation ratio 1 2(1−γ) , which is better than all previous resul...

Article history: Received 21 May 2007 Accepted 28 March 2008 Available online 4 April 2008

The purpose of this article is to explore various inequalities pertaining the numerical radius operators in a Hilbert space. Additionally, we present several bounds for Berezin number and norm that act on reproducing kernel Finally, establish necessary sufficient condition triangle inequality related hold.

The -means algorithm is by far the most widely used method for discovering clusters in data. We show how to accelerate it dramatically, while still always computing exactly the same result as the standard algorithm. The accelerated algorithm avoids unnecessary distance calculations by applying the triangle inequality in two different ways, and by keeping track of lower and upper bounds for dist...

In this paper, we deal with several reoptimization variants of the Steiner tree problem in graphs obeying a sharpened β-triangle inequality. A reoptimization algorithm exploits the knowledge of an optimal solution to a problem instance for finding good solutions for a locally modified instance. We show that, in graphs satisfying a sharpened triangle inequality (and even in graphs where edge-cos...

The study of the normability of the Lorentz spaces L(R, μ) goes back to the work of G.G. Lorentz [10, 11] (see also [13, 3, 2] for a more recent account of the normability results for the weighted Lorentz spaces). The condition defining these spaces is given in terms of the distribution function and, equivalently, the non-increasing rearrangement of f (see [1] for standard notations and basic d...

We give a short proof of the following geometric inequality: for any two triangular meshes A and B of the same polygon C, if the number of vertices in A is at most the number of vertices in B, then the maximum length of an edge in A is at least the minimum distance between two vertices in B. Here the vertices in each triangular mesh include the vertices of the polygon and possibly additional St...

Abstract In the paper, we continue research of Borsík and Doboš on functions which allow us to introduce a metric product spaces. We extend their scope broader class spaces usually fail satisfy triangle inequality, albeit they tend some weaker form this axiom. particular, examine behavior preserving b -metric inequality. provide analogues results adjusted new setting. The obtained are illustrat...

In 1964, Gähler 1 introduced the concept of 2-norm and 2-inner product spaces as generalization of norm and inner product spaces. A systematic presentation of the results related to the theory of 2-inner product spaces can be found in the book in 2, 3 and in list of references in it. Generalization of 2-inner product space for n ≥ 2 was developed by Misiak 4 in 1989. Gunawan and Mashadi 5 in 20...

We present a visual proof of a lemma that reduces the proof of the Erdős-Mordell inequality to elementary algebra. In 1935, the following problem proposal appeared in the “Advanced Problems” section of the American Mathematical Monthly [5]: 3740. Proposed by Paul Erdős, The University, Manchester, England. From a point O inside a given triangle ABC the perpendiculars OP , OQ, OR are drawn to it...