# Graphs, Dioids and Semirings

## New Models and Algorithms

• Michel Gondran
• Michel Minoux
Book

Part of the Operations Research/Computer Science Interfaces book series (ORCS, volume 41)

1. Front Matter
Pages i-xix
2. Pages 1-50
3. Pages 51-82
4. Pages 83-113
5. Pages 115-172
6. Pages 207-256
7. Pages 257-312
8. Back Matter
Pages 367-383

### Introduction

The origins of Graph Theory date back to Euler (1736) with the solution of the celebrated 'Koenigsberg Bridges Problem'; and to Hamilton with the famous 'Trip around the World' game (1859), stating for the first time a problem which, in its most recent version – the 'Traveling Salesman Problem' -, is still the subject of active research. Yet, it has been during the last fifty years or so—with the rise of the electronic computers—that Graph theory has become an indispensable discipline in terms of the number and importance of its applications across the Applied Sciences. Graph theory has been especially central to Theoretical and Algorithmic Computer Science, and Automatic Control, Systems Optimization, Economy and Operations Research, Data Analysis in the Engineering Sciences. Close connections between graphs and algebraic structures have been widely used in the analysis and implementation of efficient algorithms for many problems, for example: transportation network optimization, telecommunication network optimization and planning, optimization in scheduling and production systems, etc.

The primary objectives of GRAPHS, DIOÏDS AND SEMIRINGS: New Models and Algorithms are to emphasize the deep relations existing between the semiring and dioïd structures with graphs and their combinatorial properties, while demonstrating the modeling and problem-solving capability and flexibility of these structures. In addition the book provides an extensive overview of the mathematical properties employed by "nonclassical" algebraic structures, which either extend usual algebra (i.e., semirings), or correspond to a new branch of algebra (i.e., dioïds), apart from the classical structures of groups, rings, and fields.

### Keywords

Analysis Graph Graph theory algorithm algorithms communication computer computer science data analysis modeling operations research optimization

#### Authors and affiliations

• Michel Gondran
• 1
• Michel Minoux
• 2
1. 1.University Paris-DauphineFrance
2. 2.University Paris VIFrance