# Network Algorithms: Network Problems

## Introduction

• A network is a system of points (vertices or nodes) interconnected by lines (arcs or edges).
• Network problems are mathematical models used to represent real-life scenarios or computations involving networks.
• Types of network problems include the Travelling Salesman Problem, Shortest Path Problem, Minimum Spanning Tree Problem, and the Maximum Flow Problem.

## Shortest Path Problem

• The Shortest Path Problem seeks to find the shortest route from one point to another within a network.
• Dijkstra’s algorithm is a common method used to solve the shortest path problem.
• The algorithm works by keeping track of the shortest distance from a starting point to all other points in the network, updating these distances as it traverses the network.

## Travelling Salesman Problem

• The Travelling Salesman Problem attempts to find the shortest possible route that visits each city once and returns to the origin city.
• This is an example of a combinatorial optimization problem and is considered to be NP-hard, making it computationally challenging.

## Minimum Spanning Tree Problem

• The Minimum Spanning Tree Problem is about connecting all points in a network with the minimum total weight possible for the connecting edges.
• Two common algorithms used to solve this problem include Kruskal’s algorithm and Prim’s algorithm.
• Both algorithms use a greedy approach, choosing the smallest weight edge at each step that maintains the tree property.

## Maximum Flow Problem

• The Maximum Flow Problem deals with sending the maximum possible flow through a network, from a root (source) to a sink (target) without exceeding the capacities of the individual edges.
• The Ford-Fulkerson algorithm is typically used to solve this problem.
• This algorithm seeks to augment flow along paths from source to sink, adjusting capacities as it goes, until no more augmentations are possible.

## Advances and Applications

• Network problems and their algorithms have significant applications in societal infrastructure, such as transportation, data communication, and energy grids.
• Due to the computational complexity of many network problems, advancements in algorithm design and computational techniques are an ongoing area of research.