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Chapter 4: Problem 35

Use a greedy approach to write an algorithm for the Traveling Salesperson Problem. Show that your algorithm does not always find a minimum-length tour.

Short Answer

Expert verified
The greedy approach does not always yield the shortest possible route in the Traveling Salesperson Problem, as shown in the four-city square scenario. It can sometimes lead to unnecessarily longer routes.

Step by step solution

01

Creating the Greedy Algorithm

Start from a selected city. Continue to the nearest unvisited city until all cities have been visited. This is the Greedy Algorithm: from the current city, always go to the nearest city that has not been visited.
02

Presenting the Test Case

Consider the scenario where there are four cities, A, B, C, and D, arranged in a square, with all sides of equal length, except for the line BD which has half the length. Use the Greedy Algorithm on this configuration, beginning at city A.
03

Showing that the Algorithm Does not Find the Optimum

Starting from A, the Greedy Algorithm will proceed to whatever city is nearest. However, if it starts with either B or D, it'll eventually have to cover the longer AC line. This tour is longer compared to ABD or ACD.

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