Lower bounds for the relative greedy algorithm for approximating Steiner trees

Networks ◽  
2006 ◽  
Vol 47 (2) ◽  
pp. 111-115 ◽  
Author(s):  
Stefan Hougardy ◽  
Stefan Kirchner
Keyword(s):  
Greedy Algorithm ◽  
Lower Bounds ◽  
Steiner Trees

Lower bounds for rectilinear Steiner trees

1991 ◽  
Vol 38 (5) ◽  
pp. 279
Author(s):  
Timothy Law Snyder
Keyword(s):  
Lower Bounds ◽  
Steiner Trees

Upper and Lower Bounds for the Lengths of Steiner Trees in 3-Space

2004 ◽  
Vol 109 (1) ◽  
pp. 107-119 ◽  
Author(s):  
M. Brazil ◽  
D. A. Thomas ◽  
J. F. Weng
Keyword(s):  
Lower Bounds ◽  
Steiner Trees ◽  
Upper And Lower Bounds

scholarly journals Competitive analysis for two variants of online metric matching problem

2021 ◽  
pp. 2150156
Author(s):  
Toshiya Itoh ◽  
Shuichi Miyazaki ◽  
Makoto Satake
Keyword(s):  
Greedy Algorithm ◽  
Lower Bounds ◽  
Metric Space ◽  
Competitive Analysis ◽  
Assignment Problem ◽  
Competitive Ratio ◽  
Online Algorithm ◽  
Upper And Lower Bounds ◽  
Matching Problem ◽  
Metric Matching

In the online metric matching problem, there are servers on a given metric space and requests are given one-by-one. The task of an online algorithm is to match each request immediately and irrevocably with one of the unused servers. In this paper, we pursue competitive analysis for two variants of the online metric matching problem. The first variant is a restriction where each server is placed at one of two positions, which is denoted by OMM([Formula: see text]). We show that a simple greedy algorithm achieves the competitive ratio of 3 for OMM([Formula: see text]). We also show that this greedy algorithm is optimal by showing that the competitive ratio of any deterministic online algorithm for OMM([Formula: see text]) is at least 3. The second variant is the online facility assignment problem on a line. In this problem, the metric space is a line, the servers have capacities, and the distances between any two consecutive servers are the same. We denote this problem by OFAL([Formula: see text]), where [Formula: see text] is the number of servers. We first observe that the upper and lower bounds for OMM([Formula: see text]) also hold for OFAL([Formula: see text]), so the competitive ratio for OFAL([Formula: see text]) is exactly 3. We then show lower bounds on the competitive ratio [Formula: see text] [Formula: see text], [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text] for OFAL([Formula: see text]), OFAL([Formula: see text]) and OFAL([Formula: see text]), respectively.

An Adaptive Heuristic Approach to Compute Upper and Lower Bounds for The Close-Enough Traveling Salesman Problem

2020 ◽  
Author(s):  
Francesco Carrabs ◽  
Carmine Cerrone ◽  
Raffaele Cerulli ◽  
Bruce Golden
Keyword(s):  
Greedy Algorithm ◽  
Traveling Salesman Problem ◽  
Lower Bounds ◽  
Radio Frequency Identification ◽  
Optimal Solution ◽  
Traveling Salesman ◽  
Upper And Lower Bounds ◽  
Target Point ◽  
Research Attention ◽  
Benchmark Instances

This paper addresses the close-enough traveling salesman problem, a variant of the Euclidean traveling salesman problem, in which the traveler visits a node if it passes through the neighborhood set of that node. We apply an effective strategy to discretize the neighborhoods of the nodes and the carousel greedy algorithm to appropriately select the neighborhoods that, step by step, are added to the partial solution until a feasible solution is generated. Our heuristic, based on these ingredients, is able to compute tight upper and lower bounds on the optimal solution relatively quickly. The computational results, carried out on benchmark instances, show that our heuristic often finds the optimal solution, on the instances where it is known, and in general, the upper bounds are more accurate than those from other algorithms available in the literature. Summary of Contribution: In this paper, we focus on the close-enough traveling salesman problem. This is a problem that has attracted research attention over the last 10 years; it has numerous real-world applications. For instance, consider the task of meter reading for utility companies. Homes and businesses have meters that measure the usage of gas, water, and electricity. Each meter transmits signals that can be read by a meter reader vehicle via radio-frequency identification (RFID) technology if the distance between the meter and the reader is less than r units. Each meter plays the role of a target point and the neighborhood is a disc of radius r centered at each target point. Now, suppose the meter reader vehicle is a drone and the goal is to visit each disc while minimizing the amount of energy expended by the drone. To solve this problem, we develop a metaheuristic approach, called (lb/ub)Alg, which computes both upper and lower bounds on the optimal solution value. This metaheuristic uses an innovative discretization scheme and the Carousel Greedy algorithm to obtain high-quality solutions. On benchmark instances where the optimal solution is known, (lb/ub)Alg obtains this solution 83% of the time. Over the remaining 17% of these instances, the deviation from the optimality is 0.05%, on average. On the instances with the highest overlap ratio, (lb/ub)Alg does especially well.

Approximations and Lower Bounds for the Length of Minimal Euclidean Steiner Trees

2006 ◽  
Vol 35 (4) ◽  
pp. 573-592 ◽  
Author(s):  
J. H. Rubinstein ◽  
J. Weng ◽  
N. Wormald
Keyword(s):  
Lower Bounds ◽  
Steiner Trees

scholarly journals Lower Bounds for the Partial Grundy Number of the Lexicographic Product of Graphs

2021 ◽  
Author(s):  
Kenny Domingues ◽  
Yuri Silva de Oliveira ◽  
Ana Silva
Keyword(s):  
Greedy Algorithm ◽  
Lower Bounds ◽  
Lexicographic Product ◽  
Color Class ◽  
Grundy Number ◽  
Product Of Graphs ◽  
The Greedy Algorithm ◽  
Greedy Coloring

A Grundy coloring of a graph $G$ is a coloring obtained by applying the greedy algorithm according to some order of the vertices of $G$. The Grundy number of $G$ is then the largest $k$ such that $G$ has a greedy coloring with $k$ colors. A partial Grundy coloring is a coloring where each color class contains at least one greedily colored vertex, and the partial Grundy number of $G$ is the largest $k$ for which $G$ has a partial greedy coloring. In this article, we give some results on the partial Grundy number of the lexicographic product of graphs, drawing a parallel with known results for the Grundy number.

Sign in / Sign up

Export Citation Format

Share Document