CIRCUITS, PATHS, AND SCHEDULES Euler and Königsberg

Terminology Network – a group or system of interconnected people or things Graph – a mathematical structure consisting of vertices and edges Vertices – points on a graph that may represent locations, people, or anything of interest Edges – the lines that connect the vertices Path – a connected sequence of edges in a graph

Circuit – a path that starts and ends at the same vertex Valence – the number of edges connected to any given vertex Connected graph – for every pair of vertices there is at lest one path connecting the two vertices

Euler Circuits In an Euler circuit, you start and end at the same vertex while traveling each edge of the graph exactly once. Fact: For any given graph, it has an Euler circuit if and only if it is connected and has all vertices of even valence. If no Euler circuit exists, we may try to add edges in order to make it happen.

Eulerizations Eulerization video

Hamilton and some salesman CIRCUITS, PATHS AND SCHEDULES

W.R. Hamilton Euler focused on the traveling of edges. Hamilton was more concerned with the visiting of vertices. Hamiltonian circuit – start where you end (circuit) in which you visit each vertex once and only once. Hamiltonian circuits provide the most efficient way to run errands

Terminology Weight – a number added to an edge that may be distance, cost, or even length of time Complete graph – every pair of vertices is joined by an edge. That is, you can get to any vertex directly from the one that you are already at via only one edge.

The Best? To find the best Hamiltonian circuit for any particular weighted connected graph, you need to list all the Hamiltonian circuits and choose the shortest. This takes time and effort so being organized is key!

Traveling Salesman Problem Hamiltonian circuits led to the Traveling Salesman Problem (TSP). TSP is trying to get a salesman through his route with as little cost as possible. In order to solve the TSP we will use two types of algorithms (step-by-step procedures).

Nearest-Neighbor Algorithm Start at any vertex. Choose the next vertex by using the edge of the least weight. Choose the next vertex by using the edge of the least weight not already used, AND that does not close off the circuit. Continue until all vertices are visited. This is not always best…it is said to be a greedy algorithm.

Sorted-Edges Algorithm Make a list of the weights in an increasing order, low to high. Select an edge keeping two things in mind: We don’t have to make a circuit as we go. We cannot add an edge that would prevent a Hamiltonian circuity from being formed. Cannot use three edges from a single vertex and cannot close off a path that leaves out a vertex. This solution may not be optimal, but it is usually “good enough.”