1209 Words5 Pages

Title
Comparison of different techniques to solve the Travelling Salesperson Problem.
Abstract
In this paper, we will be explaining three ways of solving the Travelling Salesperson Problem namely the brute force method, the dynamic programming method and the branch and bound method. We will then be doing a qualitative comparison of these three methods and see which one is better for solving this problem.
Introduction
Let us first define the Travelling Salesperson Problem. The problems asks us that given a set of cities and the distances between each of the cities, which path through the cities gives us the minimum distance travelled such that each city is visited only once and at the end of the path we return to the starting city. This*…show more content…*

For a problem with n cities this approach would take us O(n!) time. We have n cities to choose from initially. (n-1) choices for the 2nd city, (n-2) choices for the third city and so on. This gives the total number of paths to be n*(n-1)*(n- 2)......(1) = n!. Hence the time taken to solve this problem will be some multiple of n!. This approach makes solving this problem even for a very small amount of cities impractical. ￼ The Dynamic Programming Method The dynamic programming method improves on the brute force method by storing the minimum cost of sub paths to avoid computing those values again and again significantly improving the time complexity. When solving a problem with dynamic programming, we define the solution with the help of a recursive relation with explains how the problem is solved step by step. For travelling salesperson problem the recursive relation to find the shortest path starting from city “1” where the set of the cities is V is as follows: G(1,V-{1}) = min { C(1,k) + G(k, V-{k,1})} where k is from the set V-{1} C(i,j) gives the cost of travelling from city i to city j. So, as explained by this recursive relation, we select a city to start with and then recursively find the distances of sub paths and keep selecting the minimum. Since we store the minimum costs of all the sub paths, the complexity of this algorithm will be the total number of sub problems that we have to solve. To find the minimum cost path starting from

For a problem with n cities this approach would take us O(n!) time. We have n cities to choose from initially. (n-1) choices for the 2nd city, (n-2) choices for the third city and so on. This gives the total number of paths to be n*(n-1)*(n- 2)......(1) = n!. Hence the time taken to solve this problem will be some multiple of n!. This approach makes solving this problem even for a very small amount of cities impractical. ￼ The Dynamic Programming Method The dynamic programming method improves on the brute force method by storing the minimum cost of sub paths to avoid computing those values again and again significantly improving the time complexity. When solving a problem with dynamic programming, we define the solution with the help of a recursive relation with explains how the problem is solved step by step. For travelling salesperson problem the recursive relation to find the shortest path starting from city “1” where the set of the cities is V is as follows: G(1,V-{1}) = min { C(1,k) + G(k, V-{k,1})} where k is from the set V-{1} C(i,j) gives the cost of travelling from city i to city j. So, as explained by this recursive relation, we select a city to start with and then recursively find the distances of sub paths and keep selecting the minimum. Since we store the minimum costs of all the sub paths, the complexity of this algorithm will be the total number of sub problems that we have to solve. To find the minimum cost path starting from

Related

## Role of Personal Selling in B2B Marketing

5193 Words | 21 Pagesdeployed to meet the needs of these segments. The salesperson augments the total product offering and serves

## Evaluation Of A Tabu Search Generator Based Memory Mechanism For Problem Solving Essay

1405 Words | 6 PagesTabu Search (TS) based memory-mechanism for problem solving. It was capable of navigating both local information

## Consumer Behavior in Tourism

2578 Words | 11 Pages1 Introduction Along with the improvement of living standard and changing consumption structure,

## Job Analysis and Recruitment Techniques Essay

1706 Words | 7 Pagesaccording his duties. Job analysis helps to solve problem in a organisation according to ability. Then they

## Compensating Sales Force

6663 Words | 27 Pagesperformance in three fundamental and interrelated ways: 1. Direct financial rewards. 2. Career advancement

## Mobile Learning

7438 Words | 30 PagesThe salesman on the way to meet a client, the worker sitting on public transport

## Case Study Of Sales Management Of Maruti Suzuki

6681 Words | 27 Pagesinformation while travelling or out of office: If an employee is not in office either travelling or on leave

## Marketing Theory and Example

7270 Words | 30 Pagespurchase on the market, or at least they can not in a way that can attract the entire purchase. Buyers are

## Introducing the History of Marketing Theory and Practice

11077 Words | 45 Pages‘marketing’ can be extended much further, all the way back to 1561 (Shaw, 1995: 16). On a related point

## Accounting Information System Chapter 1

137115 Words | 549 PagesCHAPTER 1 ACCOUNTING INFORMATION SYSTEMS: AN OVERVIEW SUGGESTED ANSWERS TO DISCUSSION QUESTIONS 1

- Conflicts, Disputes, And Solutions
- Supply Chain Technology : Rfid ( Radio Frequency Identification )
- Research Project : Drivers, Barriers And Constraints And Then Refine The Outlook Of Small Scale Power Generation
- Multinational Startup Spotify Ab Company
- How The Army 's Move A New Construct On How It Trains And Employs Brigade Combat Teams?
- The Learning Style Inventory ( Lsi )