Chapter 6: The Mathematics of Touring

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1 Traveling Salesman Problems Chapter 6: The Mathematics of Touring 6.1 What Is a Traveling Salesman Problem? The traveling salesman is a convenient metaphor for many different real-life applications. The next few examples illustrate a few of the many possible settings for a traveling salesman problem, starting, of course, with the traveling salesman s traveling salesman problem. Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: Example 6.1 A Tour of Five Cities Meet Willy, the traveling salesman. Willy has customers in five cities, which for the sake of brevity we will call A, B, C, D, and E. Willy needs to schedule a sales trip that will start and end at A (that s Willy s hometown) and goes to each of the other four cities once. We will call the trip Willy s sales tour. Other than starting and ending at A, there are no restrictions as to the sequence in which Willy s sales tour visits the other four cities. Example 6.1 A Tour of Five Cities The graph shows the cost of a one-way airline ticket between each pair of cities. Like most people, Willy hates to waste money. Thus, among the many possibilities for his sales tour, Willy wants to find the optimal (cheapest) one. How? We will return to this question soon. Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: Example 6.2 Touring the Outer Moons Example 6.2 Touring the Outer Moons It is the year An expedition to explore the outer planetary moons in our solar system is about to be launched from planet Earth. The expedition is scheduled to visit Callisto, Ganymede, Io, Mimas, and Titan (the first three are moons of Jupiter; the last two, of Saturn), collect rock samples at each, and then return to Earth with the loot. The next slide shows the mission time (in years) between any two moons. An important goal of the mission planners is to complete the mission in the least amount of time. What is the optimal (shortest) tour of the outer moons? Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e:

2 Example 6.3 Roving the Red Planet The figure shows seven locations on Mars where NASA scientists believe there is a good chance of finding evidence of life. Example 6.3 Roving the Red Planet Imagine that you are in charge of planning a samplereturn mission. First, you must land an unmanned rover in the Ares Vallis (A). Then you must direct the rover to travel to each site and collect and analyze soil samples. Finally, you must instruct the rover to return to the Ares Vallis landing site, where a return rocket will bring the best samples back to Earth. A Mars tour like this will take several years and cost several billion dollars, so good planning is critical. Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: Example 6.3 Roving the Red Planet Here are the estimated distances (in miles) that a rover would have to travel to get from one Martian site to another. What is the optimal (shortest) tour for the Mars rover? Traveling Salesman Examples In each case the problem is to find a tour of the sites (i.e., a trip that starts and ends at a designated site and visits each of the other sites once) and has the property of being optimal (i.e., has the least total cost). Any problem that shares these common elements (a traveler, a set of sites, a cost function for travel between pairs of sites, a need to tour all the sites, and a desire to minimize the total cost of the tour) is known as a traveling salesman problem, or TSP. Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: Routing School Buses Delivering Packages A school bus (the traveler) picks up children in the morning and drops them off at the end of the day at designated stops (the sites). On a typical school bus route there may be 20 to 30 such stops. With school buses, total time on the bus is always the most important variable (students have to get to school on time), and there is a known time of travel (the cost) between any two bus stops. Since children must be picked up at every bus stop, a tour of all the sites (starting and ending at the school) is required. Since the bus repeats its route every day during the school year, finding an optimal tour is crucial. Package delivery companies such as UPS and FedEx deal with TSPs on a daily basis. Each truck is a traveler that must deliver packages to a specific list of delivery destinations (the sites). The travel time between any two delivery sites (the cost) is known or can be estimated. Each day the truck must deliver to all the sites on its list (that s why sometimes you see a UPS truck delivering at 8 P.M.), so a tour is an implied part of the requirements. Since one can assume that the driver would rather be home than out delivering packages, an optimal tour is a highly desirable goal. Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e:

3 Fabricating Circuit Boards Errands Around Town In the process of fabricating integrated-circuit boards, tens of thousands of tiny holes (the sites) must be drilled in each board. This is done by using a stationary laser beam and moving the board (the traveler). To do this efficiently, the order in which the holes are drilled should be such that the entire drilling sequence (the tour) is completed in the least amount of time (optimal cost). This makes for a very high tech TSP. On a typical Saturday morning, an average Joe or Jane (the traveler) sets out to run a bunch of errands around town, visiting various sites (grocery store, hair salon, bakery, post office). When gas was cheap, time used to be the key cost variable, but with the cost of gas these days, people are more likely to be looking for the tour that minimizes the total distance traveled. Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: Modeling a TSP Modeling a TSP Every TSP can be modeled by a weighted graph, that is, a graph such that there is a number associated with each edge (called the weight of the edge). The beauty of this approach is that the model always has the same structure: The vertices of the graph are the sites of the TSP, and there is an edge between X and Y if there is a direct link for the traveler to travel from site X to site Y. Moreover, the weight of the edge XY is the cost of travel between X and Y. In this setting a tour is a Hamilton circuit of the graph, and an optimal tour is the Hamilton circuit of least total weight. In all the applications and examples we will be considering in this chapter, we will make the assumption that there is an edge connecting every pair of sites, which implies that the underlying graph model is always a complete weighted graph. The following is a summary of the preceding observations. Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: GRAPH MODEL OF A TRAVELING SALESMAN PROBLEM Sites g vertices of the graph. Costs g weights of the edges. Chapter 6: The Mathematics of Touring Tour g Hamilton circuit. Optimal tour g Hamilton circuit of least total cost 6.2 Hamilton Paths and Circuits Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e:

4 Hamilton Paths and Hamilton Circuits In Chapter 5 we discussed Euler paths and Euler circuits. There, the name of the game was to find paths or circuits that include every edge of the graph once (and only once). We are now going to discuss a seemingly related game: finding paths and circuits that include every vertex of the graph once and only once. Paths and circuits having this property are called Hamilton paths and Hamilton circuits. HAMILTON PATHS & CIRCUITS A Hamilton path in a graph is a path that includes each vertex of the graph once and only once. A Hamilton circuit is a circuit that includes each vertex of the graph once and only once. (At the end, of course, the circuit must return to the starting vertex.) Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: Euler vs. Hamilton Paths & Circuits On the surface, there is a one-word difference between Euler paths/circuits and Hamilton paths/circuits: The former covers all edges; the latter covers all vertices. But oh my, what a difference that one word makes! The figure shows a graph that (1) has Euler circuits (the vertices are all even) and (2) has Hamilton circuits. One such Hamilton circuit is A, F, B, C, G, D, E, A there are plenty more. Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: Note that if a graph has a Hamilton circuit, then it automatically has a Hamilton path the Hamilton circuit can always be truncated into a Hamilton path by dropping the last vertex of the circuit. (For example, the Hamilton circuit A, F, B, C, G, D, E, A can be truncated into the Hamilton path A, F, B, C, G, D, E.) Contrast this with the mutually exclusive relationship between Euler circuits and paths: If a graph has an Euler circuit it cannot have an Euler path and vice versa. This figure shows a graph that (1) has no Euler circuits but does have Euler paths (for example C, D, E, B, A, D) and (2) has no Hamilton circuits (sooner or later you have to go to C, and then you are stuck) but does have Hamilton paths (for example, A, B, E, D, C). This illustrates that a graph can have a Hamilton path but no Hamilton circuit! Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e:

5 This figure shows a graph that (1) has neither Euler circuits nor paths (it has four odd vertices) and (2) has Hamilton circuits (for example A, B, C, D, E, A there are plenty more) and consequently has Hamilton paths (for example, A, B, C, D, E). This figure shows a graph that (1) has Euler circuits (the vertices are all even) and (2) has no Hamilton circuits (no matter what, you are going to have to go through E more than once!) but has Hamilton paths (for example, A, B, E, C, D). Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: This figure shows a graph that (1) has no Euler circuits but has Euler paths (F and G are the two odd vertices) and (2) has neither Hamilton circuits nor Hamilton paths. This figure shows a graph that (1) has neither Euler circuits nor Euler paths (too many odd vertices) and (2) has neither Hamilton circuits nor Hamilton paths. Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: Summary Euler versus Hamilton The lesson of is that the existence of an Euler path or circuit in a graph tells us nothing about the existence of a Hamilton path or circuit in that graph. This is important because it implies that Euler s circuit and path theorems from Chapter 5 are useless when it comes to identifying Hamilton circuits and paths. But surely, there must be analogous Hamilton circuit and path theorems that we could use to determine if a graph has a Hamilton circuit, a Hamilton path, or neither. Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e:

6 Euler versus Hamilton Surprisingly, no such theorems exist. Determining when a given graph does or does not have a Hamilton circuit or path can be very easy, but it also can be very hard it all depends on the graph. Copyright Pearson Education, Education. Inc. All rights reserved. Excursions in Modern Mathematics, 7e: