Excursions in Modern Mathematics(Tannenbaum) and Thinking Mathematically (Blitzer)

A C B D Steps: 1.List numbers in order from least to greatest. 2.Check number from smallest to largest if the circuit allows you to follow the path. 3.Add together all the checked numbers.

Is performed by continually taking an edge with the smallest weight. This will give you approximate solutions to traveling salesperson problems.

1. Start at the designated starting vertex. If there is no designated staring vertex pick any vertex. 2. From the starting vertex go to its nearest neighbor (i.e. the vertex for which the corresponding edge has the smallest weight). 3. From each vertex go to the nearest neighbor, choosing only among the vertices that have not been visited (if there are more than one nearest neighbor with the same measure choose from them at random). Continue the process until all the vertices have been visited. 4. From the last vertex return to the starting vertex.

Is similar to nearest neighbor but you choose which vertex to start from. Repeat the process several times, each time starting from a different vertex. You may find different “nearest neighbor solutions” Then you can pick the best.

You use this process to piece together a tour by picking the separate “links” of the tour on the basis of cost. Steps: 1. Pick the cheapest length(i.e. the edge with the smallest weight) available. (In case of a tie pick one at random.) Mark it(say in red). 2. Pick the next cheapest link available and mark it. 3,4,…,n-1. Continue picking and marking the cheapest length unmarked link available that does not: a. Close a circuit. b. Create three edges coming out of a single vertex. 3. Connect the last two vertices to close the red circuit. This circuit gives us the Cheapest-link tour.