# Network Algorithms: Least Weight Route Through All Vertices That Traverses Every Arc At Least Once

## Introduction

• This topic is related to network algorithms within discrete mathematics.
• It specifically deals with the tasks of finding the least weight route that visits all vertices and traverses each arc at a minimum of once.
• This is a frequent problem in the sphere of graph theory and network analysis.
• Algorithms commonly employed for these tasks include the Chinese Postman Problem and Euler’s Theorem.

## Chinese Postman Problem

• The Chinese Postman Problem (also known as the Route Inspection Problem) is an algorithm used to find the shortest path that will visit every arc of a graph at least once and return to the starting vertex.
• It can be applied to both directed and undirected graphs.
• If the graph is undirected and has more than two vertices of odd degree, or if it is directed and has more than two vertices where the in-degree and out-degree differ, then it doesn’t have an Eulerian circuit, and we have to duplicate some edges to create one.
• The optimal solution of the Chinese Postman Problem is an Euler Tour augmented with the shortest path between unmatched vertices.

## Euler’s Theorem

• Euler’s Theorem also provides a solution for the least weight route problem.
• It states that a connected graph has an Euler Circuit (a route that traverses every edge once and returns to its start point) if and only if the degree of each vertex is even for an undirected graph, or each vertex has equal in-degree and out-degree for a directed graph.
• An Euler Path (traverses every edge once but does not necessarily return to the start point) exists if the graph is connected and has exactly two vertices of odd degree.

## Real World Applications

• These network algorithms play a significant role in many fields including logistics, transportation planning, telecommunications and computer networks.
• They are used whenever one needs to find the most efficient route to traverse a given network, for example, when planning postal delivery routes, garbage collection routes, or for routing data in a network.

## Examples

• For instance, consider a delivery network with various locations (vertices) and paths of various costs (arcs) among them. The task is to find the most cost-efficient route that goes through all routes at least once and then returns to the initial location. In this case, the delivery network becomes a graph, and the Chinese Postman Problem or Euler’s theorem can be applied to find the least weight route.
• In another scenario, suppose you are trying to design a walking tour that includes all tourist attractions in a city. The locations can be modelled as vertices, the paths between them as arcs, and the distances or time taken to traverse them as weights. Using these algorithms, you can find the tour that visits each location once and minimises total distance or time.