scholarly journals Conceptual Framework for Finding Approximations to Minimum Weight Triangulation and Traveling Salesman Problem of Planar Point Sets

2020 ◽  
Vol 11 (4) ◽  
Author(s):  
Marko Dodig ◽  
Milton Smith
Keyword(s):  
Conceptual Framework ◽  
Traveling Salesman Problem ◽  
Minimum Weight ◽  
Traveling Salesman ◽  
Point Sets ◽  
Planar Point ◽  
Minimum Weight Triangulation

DIAMONDS ARE NOT A MINIMUM WEIGHT TRIANGULATION'S BEST FRIEND

2002 ◽  
Vol 12 (06) ◽  
pp. 445-453 ◽  
Author(s):  
PROSENJIT BOSE ◽  
LUC DEVROYE ◽  
WILLIAM EVANS
Keyword(s):  
Polynomial Time ◽  
Minimum Weight ◽  
Connected Subgraph ◽  
Constant Number ◽  
Point Sets ◽  
Expected Number ◽  
Number Of Components ◽  
Best Friend ◽  
Point Set ◽  
Minimum Weight Triangulation

Two recent methods have increased hopes of finding a polynomial time solution to the problem of computing the minimum weight triangulation of a set S of n points in the plane. Both involve computing what was believed to be a connected or nearly connected subgraph of the minimum weight triangulation, and then completing the triangulation optimally. The first method uses the light graph of S as its initial subgraph. The second method uses the LMT-skeleton of S. Both methods rely, for their polynomial time bound, on the initial subgraphs having only a constant number of components. Experiments performed by the authors of these methods seemed to confirm that randomly chosen point sets displayed this desired property. We show that there exist point sets where the number of components is linear in n. In fact, the expected number of components in either graph on a randomly chosen point set is linear in n, and the probability of the number of components exceeding some constant times n tends to one.

Almost empty polygons

2003 ◽  
Vol 40 (3) ◽  
pp. 269-286 ◽  
Author(s):  
H. Nyklová
Keyword(s):  
Exact Results ◽  
Point Sets ◽  
Planar Point ◽  
Convex Position ◽  
Vertex Set ◽  
Point Set ◽  
Empty Polygons

In this paper we study a problem related to the classical Erdos--Szekeres Theorem on finding points in convex position in planar point sets. We study for which n and k there exists a number h(n,k) such that in every planar point set X of size h(n,k) or larger, no three points on a line, we can find n points forming a vertex set of a convex n-gon with at most k points of X in its interior. Recall that h(n,0) does not exist for n = 7 by a result of Horton. In this paper we prove the following results. First, using Horton's construction with no empty 7-gon we obtain that h(n,k) does not exist for k = 2(n+6)/4-n-3. Then we give some exact results for convex hexagons: every point set containing a convex hexagon contains a convex hexagon with at most seven points inside it, and any such set of at least 19 points contains a convex hexagon with at most five points inside it.

Grafos hamiltonianos aplicado al turismo de Panamá

2019 ◽  
Vol 7 (1) ◽  
pp. 109-113
Author(s):  
Julio Trujillo
Keyword(s):  
Traveling Salesman Problem ◽  
Traveling Salesman

Un problema clásico de Teoría de Grafos es encontrar un camino que pase por varios puntos, sólo una vez, empezando y terminando en un lugar (camino hamiltoniano). Al agregar la condición de que sea la ruta más corta, el problema se convierte uno de tipo TSP (Traveling Salesman Problem). En este trabajo nos centraremos en un problema de tour turístico por la ciudad de Panamá, transformándolo a un problema de grafo de tal manera que represente la situación planteada.

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