MATH 103: Contemporary Mathematics Study Guide for Chapter 6: Hamilton Circuits and the TSP

Transcription

1 MTH 3: ontemporary Mathematics Study Guide for hapter 6: Hamilton ircuits and the TSP. nswer the questions above each of the following graphs: (a) ind 3 different Hamilton circuits for the graph below. D G (b) Practice a) the rute orce lgorithm, b) the Nearest Neighbor lgorithm, c) the Repeatitive Nearest Neighbor lgorithm, and d) the heapest Link lgorithm for the two graphs below: 275 D I H G (c) ind a solution of the TSP problem for the following graph. (areful! the graph is not complete) D 2. Go back to page 68 of hapter 5 and read the definition of adjacent vertices. Sometimes students use the word connected when they mean adjacent. e careful of this point in the discussion questions. 3. xplain the difference between an uler circuit and a Hamilton circuit.

2 4. (a) Within a Hamilton circuit (considered as a graph on its own), every vertex has degree (b) If a graph has a vertex of degree one, can this graph contain a Hamilton circuit? xplain. (c) If a graph contains a Hamilton circuit, then every vertex of the graph has degree at least (d) If every vertex of a connected graph has degree two or more, does the graph necessarily have a Hamilton circuit? xplain. 5. If you are having trouble remembering the difference between an uler circuit and a Hamilton circuit, you may want to recall some of the actual applications in each case. or example, police patrol and garbage collection routes are modelled by uler ircuits. Delivery problems, such as the one of a travelling salesman, are modelled by Hamilton ircuits. xplain why these two types of problems require different graph theory models. 6. Practice finding Hamilton circuits and computing their total weight by working problems 7,4,5 on pages What is the Travelling Salesman Problem? 8. Practice the Nearest Neighbor lgorithm for the graph below. Use as your starting vertex. Indicate your answer as a list of vertices and compute its total weight. Then practice the Repeatitive Nearest Neighbor (review how it works first!): is the solution provided by it a better one? D

3 9. Use the heapest Link lgorithm to find a Hamilton circuit for the following weighted graph. Show your work (you can list the edges in the order you choose them). t the end describe your circuit as a list of vertices in the order in which you visit them. re you allowed to visit any vertex more than once? ompute the weight of the circuit D G. Place weights (of your choice) on each edge of the following graph. Then use any method to find a Hamilton circuit. Highlight your Hamilton circuit on the graph.. Provide an example of a graph which has a Hamilton circuit (come up with your own example, do not copy it from the book or the notes). Highlight the Hamilton circuit in your graph. 2. Which algorithms of hapter 3 (rute orce, heapest Link, Nearest Neighbor, Repeatitive Nearest Neighbor) produce optimal (minimal weight) Hamilton circuits? Which produce Hamilton circuits which are (in most cases) only close to optimal? 3. If a graph has a Hamilton circuit, then the rute orce algorithm will always yield an optimal Hamilton circuit. Why?

4 4. Read about the rute orce algorithm and the section at pages 2-2 to appreciate why the rute orce algorithm is not an efficient method of solving the travelling salesman problem with twenty or more cities (vertices). 5. Practice the rute orce algorithm showing complete work (list of Hamilton circuits, computation of total weight, etc.) for the graph in problem number 7 of this study guide. 6. or the Nearest Neighbor lgorithm, are we allowed to visit a vertex (other than the starting vertex) twice? Why or why not? 7. o the heapest Link algorithm, we discard an edge if marking it would lead to three marked edges out of a single vertex. Why? 8. (More difficult question:) or the heapest Link algorithm, we discard an edge if marking it would lead to a marked circuit which does not include all vertices. Why? (areful: the sentence we want a circuit which includes all vertices is not a complete answer). 9. In the following exercises the words complete, connected, and adjacent are crucial. Review their definitions before answering the following questions. (a) re two adjacent vertices necessarily connected? (b) re two connected vertices necessarily adjacent? (c) graph is connected if any two vertices are. (d) graph is complete if any two vertices are. 2. complete graph with three or more vertices always has a Hamilton circuit. How do we know this?

5 2. In a complete graph with N vertices there are edges, Hamilton circuits starting at a given vertex, distinct Hamilton circuits starting at a given vertex. arefully explain how we derived each of the formulas. 22. Draw a connected graph with six vertices and eighteen edges. Does it have a Hamilton circuit? If not, then add a few edges so it does. Next, randomly number its edges starting from. Now, try using the Nearest Neighbor algorithm to find a Hamilton circuit in your weighted graph. Did the algorithm work? 23. Is the graph above complete? 24. an you have a Hamilton circuit in a graph which is not complete? 25. xplain why the rute orce lgorithm is not an efficient algorithm. What happens to the number of operations (e.g. the number of Hamilton ircuits you need to check) as you increase the number of vertices of your complete graph from N to N+? 26. Think about applying the Nearest Neighbor or the heapest Link algorithms to a connected weighted graph, focusing on the last edge which you mark. oth algorithms will yield a Hamilton circuit for every complete graph. (a) Will these algorithms (Nearest Neighbor or heapest Link) yield a Hamilton circuit for every connected weighted graph? Why or why not? (b) Will these algorithms yield a Hamilton circuit for every connected weighted graph which has a Hamilton circuit? In other words, suppose someone constructs a weighted graph which has a Hamilton circuit. Will the two algorithms be guaranteed to find Hamilton(s) circuits in this graph? xplain.

6 (c) Will the heapest Link algorithm yield a Hamilton circuit for some connected graphs which are not complete? Why or why not? (d) Why does the heapest Link (or Nearest Neighbor/Repeatitive Nearest Neighbor) yield a Hamilton circuit for every complete weighted graph with three or more vertices? (note your answer to part (b) and be careful; it is not enough to know that a graph has a Hamilton circuit). 27. Why are the Nearest Neighbor, Repetitive Nearest Neighbor, and heapest Link algorithms called efficient? What happens to the number of operations (say the number of weights you have to add up to get the total weight of the resulting Hamilton circuit) as you increase increase the number of vertices of your complete graph from N to N+? 28. an a circuit contain another smaller subcircuit? an a Hamilton circuit contain a smaller subcircuit? 29. onsider a graph with N vertices. xplain why the following statement is true: If this graph has a Hamilton circuit, then such a circuit must contain N edges of the graph. 3. onstruct an example of a complete weighted graph for which the heapest Link algorithm produces the worst possible Hamilton circuit (i.e. the one with the highest total weight). 3. onstruct an example of a complete weighted graph (different from the previous one) for which the Nearest Neighbor algorithm produces the worst possible Hamilton circuit (i.e. the one with the highest total weight). 32. Why are the Nearest Neighbor, Repetitive Nearest Neighbor, and heapest Link algorithms for solving the TSP called non-optimal?