What is Graph Theory, and why should you care?

What is Graph Theory, and why should you care?

What is Graph Theory, and why should you care? From graph theory to path optimization Graph theory might sound like an intimidating and abstract topic to you, so why

From graph theory to path optimization

Graph theory might sound like an intimidating and abstract topic to you, so why should you even spend your time reading an article about it? However, although it might not sound very applicable, there are actually an abundance of useful and important applications of graph theory! In this article, I will try to explain briefly what some of these applications are. In doing so, I will do my best to convince you that having at least some basic knowledge of this topic can be useful in solving some interesting problems you might come across.

In this article, I will through a concrete example show how a route planning/optimization task can be formulated and solved using graph theory. More specifically, I will consider a large warehouse consisting of 1000s of different items in various locations/pickup points. The challenge here is, given a list of items, which path should you follow through the warehouse to pickup all items, but at the same time minimize the total distance traveled? For those of you familiar with these kind of problems, this has quite some resemblance to the famous traveling salesman problem. (A well known problem in combinatorial optimization, important in theoretical computer science and operations research).

As you might have realized, the goal of this article is not to give a comprehensive introduction to graph theory (which would be quite a tremendous task). Through a real-world example, I will rather try to convince you that knowing at least some basics of graph theory can prove to be very useful!

I will start with a brief historical introduction to the field of graph theory, and highlight the importance and the wide range of useful applications in many vastly different fields. Following this more general introduction, I will then shift focus to the warehouse optimization example discussed above.

The history of Graph Theory

The basic idea of graphs were first introduced in the 18th century by the Swiss mathematician Leonhard Euler, one of the most eminent mathematicians of the 18th century (and of all time, really). His work on the famous “Seven Bridges of Königsberg problem”, are commonly quoted as origin of graph theory.

The city of Königsberg in Prussia (now Kaliningrad, Russia) was set on both sides of the Pregel River, and included two large islands — Kneiphof and Lomse — which were connected to each other, or to the two mainland portions of the city, by seven bridges (as illustrated in the below figure to the left). The problem was to devise a walk through the city that would cross each of those bridges once and only once.

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