1. The Highway Inspector s Problem

Transcription

1 MATH 100 Survey of Mathematics Fall The Highway Inspector s Problem The Königsberg Bridges over the river Pregel C c d e A g D a B b Figure 1. Bridges f Is there a path that crosses every bridge exactly once? Leonhard Euler ( )

2 2 Abstraction and mathematical modeling. What is relevant to the question? Which features can be ignored in order to simplify the problem? C c d e A D a b f B Figure 2. Euler s Model Result. An Euler path has one starting point and one terminal point. All four points on Euler s graph must be starting points or end points of an Euler path. This is impossible. There is no Euler path.

3 A new field is born: Graph Theory. 3 What is graph theory? Describe and classify all graphs (too hard). Which important properties do graphs have? (degree, connectivity, completeness, loops) Describe and classify certain graphs. Graphsareusedtopresentdatainanefficienteasytoread fashion: Organizational chart, networks, scheduling multitasking, precedence graphs, storing and retrieving data. Definitions. A graph consists of a(finite) set of points, called vertices and certain lines between vertices called edges. Definition. The degree of a vertex in a graph is the number of vertices ending (or starting) at the vertex.

4 4 2. The Traveling Salesman Problem The problem of the highway inspector. The highway inspector wishes to traverse every highway exactly once. The problem of the traveling salesman. A salesman wishes to visit the towns of his district in such a way that he visitseachtownexactlyonce. Canhedoit? Howwillhechoose his route? Definition 2.1. A Hamilton path 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 except that starting point and end point coincide. Proposition 2.2. The highway inspector s problem and the traveling salesman s problem are different. Proof. Example 6.1. Remark 2.3. While the highway inspector s problem has a nice and easy solution, the traveling salesman s problem is unsolved. Therefore, consider special cases. Hunch: The more roads, the better the chance for the traveling salesman.

5 Definition 2.4. A graph with N vertices is complete if every two distinct vertices are joint by an edge. Notation: K N. Theorem 2.5. (1) The complete graph K N has N(N 1)/2 edges. (2) Of all graphs with N vertices that have no loops or multiple edges, the complete graph has the most edges. Theorem 2.6. There are (N 1)! distinct Hamilton circuits in K N. Dirac s Theorem. Suppose that the district of the traveling salesman contains at least three towns and let N be the number of towns. If every town has at least N/2 streets ending in it, then there is a route for the traveling salesman. 5 Portland Boise Butte Sacramento Reno Salt Lake City Figure 3

6 6 Figure 4 The extended problem of the traveling salesman. A salesman wishes to visit the towns of his district in such a way thathevisitseachtownandtheexpenseofhispathisassmall as possible. Which path should he choose? The graph model of the extended problem of the traveling salesman. Agraphisweightedifeachedgeisassignedacost,its weight. AHamiltoncircuitisatourandanoptimal tourisahamilton circuit of least weight. Applications. (Book pp ) Tour of five cities. Touring the outer moons. Roving the Red Planet. Touting school buses. Delivering packages. Fabricating circuit boards. Running errands around town.

7 The diagram below shows a weighted simple graph. Look at the weight as distances. What is the shortest path from a to z? b 3 c a 1 z 2 1 d 4 Figure 5 e

8 8 Strategies Exhaustive search. Take the cheapest next leg. Algorithms Brute force algorithm. Optimal but inefficient. Check all possibilities. Choose one of the cheapest. Nearest-neighbor algorithm. Approximate. Choose a vertex X. Find nearest neighbor tour. Repetitive nearest-neighbor algorithm. Approximate. For every vertex X, find nearest neighbor tour and compute the cost. Choose the circuit of least cost. Cheapest link algorithm. Choose cheapest edge. Mark. Repeatchoosingthenextcheapestedgeandmarkprovided (1) it does not close a circuit (2) it does not add a third edge two a vertex with two edges marked. Connect the last two vertices.