Q1a In every instance (i.e., example) of the TSP, we are given n cities, where each pair of cities is connected by a weighted edge that measures the cost of traveling between those two cities. Our goal is to find the optimal TSP tour, minimizing the total cost of a Hamiltonian cycle in G. Although it is NP-complete to solve the TSP, there is a simple 2-approximation achieved by first generating a minimum-weight spanning tree of G and using this output to determine our TSP tour. Prove that our output is guaranteed to be a 2-approximation, provided the Triangle Inequality holds. In other words, if OPT is the total cost of the optimal solution, and APP is the total cost of our approximate solution, clearly explain why APP ≤ 2* OPT. 1b Let G be this weighted undirected graph, containing 7 vertices and 11 edges. A 5 D 7 9 6 B 15 F 8 7 8 11 5 E 9 с G For each of the 10 edges that do not appear (AC, AE, AF, AG, BF, BG,CD,CF,CG, DG), assign a weight of 1000. It is easy to see that the optimal TSP tour has total cost 51.

Q1a In every instance (i.e., example) of the TSP, we are given n cities, where each pair of cities is connected by a weighted edge that measures the cost of traveling between those two cities. Our goal is to find the optimal TSP tour, minimizing the total cost of a Hamiltonian cycle in G. Although it is NP-complete to solve the TSP, there is a simple 2-approximation achieved by first generating a minimum-weight spanning tree of G and using this output to determine our TSP tour. Prove that our output is guaranteed to be a 2-approximation, provided the Triangle Inequality holds. In other words, if OPT is the total cost of the optimal solution, and APP is the total cost of our approximate solution, clearly explain why APP ≤ 2* OPT. 1b Let G be this weighted undirected graph, containing 7 vertices and 11 edges. A 5 D 7 9 6 B 15 F 8 7 8 11 5 E 9 с G For each of the 10 edges that do not appear (AC, AE, AF, AG, BF, BG,CD,CF,CG, DG), assign a weight of 1000. It is easy to see that the optimal TSP tour has total cost 51.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Concept explainers

Duality

Duality is a linear programming (LP) concept which states that it is possible to interpret every LP problem using two distinct methods, but they will have similar solutions. That means any LP problem can be mentioned in another equivalent form based on identical data.

Question

Q1a
In every instance (i.e., example) of the TSP, we are given n cities, where each pair of cities is
connected by a weighted edge that measures the cost of traveling between those two cities.
Our goal is to find the optimal TSP tour, minimizing the total cost of a Hamiltonian cycle in G.
Although it is NP-complete to solve the TSP, there is a simple 2-approximation achieved by
first generating a minimum-weight spanning tree of G and using this output to determine our
TSP tour.
Prove that our output is guaranteed to be a 2-approximation, provided the Triangle Inequality
holds. In other words, if OPT is the total cost of the optimal solution, and APP is the total cost
of our approximate solution, clearly explain why APP ≤ 2* OPT.
1b
Let G be this weighted undirected graph, containing 7 vertices and 11 edges.
|
A
5
D
1
9
6
B
15
F
8
7
8
11
5
E
9
с
G
For each of the 10 edges that do not appear
(AC, AE, AF, AG, BF, BG, CD, CF, CG, DG), assign a weight of 1000. It is easy to
see that the optimal TSP tour has total cost 51.
Generate an approximate TSP tour using the algorithm from Question 1a, and
calculate the total cost of your solution. Explain why your solution is not a 2-
approximation

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