Spring 2015Mathematics in

Management Science

Network Problems

Networks & TreesMinimum NetworksSpanning TreesMinimum Spanning Trees

TreesA tree is a connected graph with no circuits.

A graph is a tree if and only if there is a

unique path connecting any two vertices.

Networks & Trees

A network is a connected graph.

A tree is a network which contains no circuits.

Given a network, we’ll search for subgraphs which are

spanning trees.

Trees & Spanning Trees

A tree is a network (a connected graph) which contains no circuits.

We’ll look for subgraphs of a given network which are trees.

Such a tree which contains all of the vertices of original network is called a spanning tree.

Spanning Trees

A spanning tree of a graph:• is a subgraph (uses only edges &

vertices of the original graph)• is a tree (no loops)• includes all original vtxs (it spans)

Spanning Trees

The network in (a) has 8 edges, 5 of which are bridges.

There are 3 different spanning trees pictured in (b), (c), and (d).

Minimal Cost Spanning Trees

Of all possible spanning trees, the one with the smallest total weight is called a

minimum spanning tree.First Network Problem Given weighted network, find

a minimum spanning tree.

First Network Problem

Given weighted network, find a minimum spanning tree.

Any spanning tree solves connectivity problem. However, some connections cost more than others!

Want to find a spanning tree of least cost.

Use Kruskal’s Algorithm.

Kruskal’s Algorithm

Add edges, in order of increasing cost, so that

• no circuits get formed• all vertices on some added edge

(when done)• connected (when done)

Example

Find the minimal cost spanning tree.

Example

Find the minimal cost spanning tree.

Optimality of Kruskal’s Algorithm

Kruskal’s Algorithm works for any network, is fast & greedy, and always yields an optimal solution to the Minimum Spanning Tree Problem.

In contrast, the Sorted Edges Algorithm (used to attack the Traveling Salesman Problem) does not always yield the optimal solution.

How to tell when you are done?

When all vtxs included and connected.

If adding any new edge creates circuit (so all remaining edges rejected).

For n vertices, a spanning tree always has exactly (n − 1) edges.

⇒ If used (n − 1) edges and no “loops”, then must have spanning tree!

Example

Which would most likely play a role in solving each problem:• an Euler circuit ?• a Hamiltonian circuit ?• A spanning tree ?

A Brazilian company needs to run telephone cable to a remote villages in the Amazon. How should it do this in order to use the least amount of cable?

Example

Which would most likely play a role in solving each problem:• an Euler circuit ?• a Hamiltonian circuit ?• A spanning tree ?

A limousine company needs to take a businessmen around to visit several potential development sites. What route will minimize the driving distance?

Example

Which would most likely play a role in solving each problem:• an Euler circuit ?• a Hamiltonian circuit ?• A spanning tree ?

A candidate in upcoming election wants to deliver campaign literature to each house in her ward. How can she lay out her route as efficiently as possible?