Chapter 14.3

minimum-cost Hamiltonian circuit

A Hamiltonian circuit in a graph with weights on the edges, for which the sum of the weights of the edges of the Hamiltonian circuit is as small as possible

A path that passes through all the vertices of a graph exactly once is called a _____________ path.

Hamilton

Such a path that begins and ends at the same vertex and passes through all other vertices exactly once is called a _________________ circuit.

Hamilton

A complete graph with three or more vertices will have more than one

Hamilton circuit

Hamilton circuit

a circuit that visits each vertex once and only once. (Remember that circuits start and end at the same vertex, so the beginning of the vertex is also listed at the end.) In the image below, A,C,D,E,F,B,A is a Hamiltonian circuit and A,B,F,E,D,C,A is one as well.

If such a graph has n​ vertices, the number of Hamilton circuits in the graph is given by the factorial expression​ ________.

(n-1)!

A method that approximates the solution to the salesperson problem is called the​ ______ Method. This method involves continually choosing an edge with the smallest​ ______.

1. Nearest Neighbor Method 2. weight

Each starting vertex would yield one Hamilton circuit, so if you had 10 vertices you would have...

10 Hamilton circuits

One such category where every graph has a Hamilton circuit is

complete graphs

Brute-Force Method

construct a tree diagram showing all possible Hamilton circuits.

true or false The Nearest Neighbor Method provides exact solutions to the traveling salesperson problems.

The statement is false. The Nearest Neighbor Method is meant only to approximate the optimal solution.

true or false In a weighted​ graph, the lengths of the edges are proportional to their weights.

The statement is​ false, the lengths of the edges need not represent​ real-world values and are often changed to make a graph more easily read.

complete graph

a graph where every pair of vertices is joined by an edge.

weighted graph

a graph whose edges all have numbers attached to them.

weight

a number assigned to an edge that usually represents a cost, a distance, or a time.

Hamilton path

a path that passes through each vertex of a graph exactly once.

algorithm

a step-by-step process used to solve a problem that often consists of repeating a part of the process a finite number of times until the end result is reached.

On a pencil and paper graph, a Hamiltonian circuit and Hamilton path can be indicated by...

a wiggled line and by listing the vertices from the starting vertex through each visited vertex and back to the same starting vertex. On MyMathLab a Hamilton circuit and Hamilton path are indicated by listing the vertices in the order they are traversed with commas separating the vertices in the listing.

two methods used to solve traveling salesperson problems

brute-force method & nearest-neighbor method

A _______________ circut a complete graph that contains three or more vertices.

hamilton

The nearest-neighbor method falls into a class of algorithms called

heuristic algorithms

The problem of finding a Hamilton circuit for which the sum of these numbers is a minimum is called the traveling salesperson problem. Such a Hamilton circuit is called the _________________ solution.

optimal