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Chapter 10: Problem 28

Find a route with the least total airfare that visits each of the cities in this graph, where the weight on an edge is the least price available for a flight between the two cities.

Short Answer

Expert verified
Generate and compare the total airfare of all routes; the one with the minimum total cost is the best.

Step by step solution

01

Understand the Problem

The problem is asking for the route with the least total airfare that visits every city exactly once. This is essentially the Traveling Salesman Problem given airfare costs as edge weights.
02

List All Cities and Flights

Write down the names of all the cities and the flight costs between each pair of cities. Represent this information in a matrix or list form.
03

Generate Permutations

List all possible permutations of the cities. Each permutation represents a potential route.
04

Calculate Total Airfare for Each Route

For each permutation, calculate the total airfare by summing up the weights of the edges (flight costs) between the cities in that order.
05

Identify the Minimum Total Airfare

Compare the total airfare of all the routes. The route with the smallest total airfare is the optimal route.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graph Theory
Graph theory is a field of mathematics that deals with the study of graphs. A graph consists of vertices (also called nodes) and edges, which connect pairs of vertices. In the context of the Traveling Salesman Problem (TSP), each city is a vertex, and the routes (or flights) between cities are the edges.

To visualize the problem, think of a map where each city is a point. Draw lines between these points to represent flights. If you assign a number to each line (representing airfare), your graph now has weighted edges. These weights are crucial in solving TSP, as they represent the cost of traveling from one city to another.
Minimum Cost Route
The goal of the Traveling Salesman Problem is to find a route that visits each city exactly once and has the lowest possible total cost. This is known as finding the minimum cost route.

Here's what you need to do:
  • Identify all potential routes that visit each city once.
  • Calculate the total cost for each route by adding up the airfare prices between consecutive cities.
  • Compare these costs and identify the route with the smallest total airfare.

    • To illustrate, if there are three cities A, B, and C, you must check routes like A-B-C, B-C-A, and C-A-B. Calculate the total airfare for each and choose the one with the lowest cost.
Permutations
In the Traveling Salesman Problem, permutations play a key role in exploring all possible routes. A permutation is an ordered arrangement of items. For example, with cities A, B, and C, possible permutations are ABC, ACB, BAC, BCA, CAB, and CBA.

By listing all permutations, you ensure that no potential route is overlooked. This can be done by:
  • Writing down all the cities.
  • Generating all possible orders in which to visit them.

    • Each permutation represents a different route through the cities. You will then calculate the total airfare for each permutation to find the optimal (cheapest) route.
Edge Weights
Edge weights are numbers assigned to the edges of the graph that indicate the cost or distance between two vertices. In the Traveling Salesman Problem, these weights represent the airfare between cities.

To proceed, you must:
  • List all the flights and their respective costs.
  • Create a matrix or a list to organize this information.
  • Use these weights to calculate the total cost of each potential route.

    • Think of the edge weights as key information that will help you evaluate and compare different routes. By summing up these weights for each permutation, you can find the route with the minimum airfare.

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Most popular questions from this chapter

The parts of this exercise outline a proof of Ore's theorem. Suppose that \(G\) is a simple graph with \(n\) vertices, \(n \geq 3,\) and \(\operatorname{deg}(x)+\operatorname{deg}(y) \geq n\) whenever \(x\) and \(y\) are non-adjacent vertices in \(G .\) Ore's theorem states that under these conditions, \(G\) has a Hamilton circuit. a) Show that if \(G\) does not have a Hamilton circuit, then there exists another graph \(H\) with the same vertices as \(G,\) which can be constructed by adding edges to \(G,\) such that the addition of a single edge would produce a Hamilton circuit in \(H .[\text {Hint} : \text { Add as many edges as }\) possible at each successive vertex of \(G\) without producing a Hamilton circuit.] b) Show that there is a Hamilton path in \(H\) c) Let \(v_{1}, v_{2}, \ldots, v_{n}\) be a Hamilton path in \(H .\) Show that \(\operatorname{deg}\left(v_{1}\right)+\operatorname{deg}\left(v_{n}\right) \geq n\) and that there are at most \(\operatorname{deg}\left(v_{1}\right)\) vertices not adjacent to \(v_{n}\) (including \(v_{n}\) itself). d) Let \(S\) be the set of vertices preceding each vertex adjacent to \(v_{1}\) in the Hamilton path. Show that \(S\) contains \(\operatorname{deg}\left(v_{1}\right)\) vertices and \(v_{n} \notin S .\) e) Show that \(S\) contains a vertex \(v_{k}\) that is adjacent to \(v_{n}\) implying that there are edges connecting \(v_{1}\) and \(v_{k+1}\) and \(v_{k}\) and \(v_{n} .\) f) Show that part (e) implies that \(v_{1}, v_{2}, \ldots, v_{k-1}\) \(v_{k}, v_{n}, v_{n-1}, \ldots, v_{k+1}, v_{1}\) is a Hamilton circuit in \(G\) . Conclude from this contradiction that Ore's theorem holds.

Show that the property that a graph is bipartite is an isomorphic invariant.

Is a shortest path between two vertices in a weighted graph unique if the weights of edges are distinct?

Show that a bipartite graph with an odd number of vertices does not have a Hamilton circuit.

Show that a directed multigraph having no isolated vertices has an Euler path but not an Euler circuit if and only if the graph is weakly connected and the in-degree and out-degree of each vertex are equal for all but two vertices, one that has in-degree one larger than its out- degree and the other that has out-degree one larger than its in-degree.
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