Earlier this semester an OU math professor, Dr. Murad Özaydin, and a former OU grad student, Dr. James Dover (now at Cameron University in Lawton), posted their most recent paper on the ArXiv.

It is entitled “Homeomorphism and Homotopy Types of Restricted Configuration Spaces of Metric Graphs”. But really, as Dr. Özaydin explained in the topology seminar, it’s about Robots on Rails.

As part of our very occasional postings on research in the OU Math department, we wanted to tell you about their very cool theorem. Before we do that, let’s talk robots.

In real life, robots which can move are often only allowed to move along certain paths. They might actually move along rails; or, even if they just drive along on wheels in a factory their programming steers them along certain fixed paths so they don’t accidentally run into walls, expensive machines, people, etc.

And even non-robots often travel on real or virtual rails. Trains, of course. But cars on the freeway system, planes flying through the air (commercial planes can’t go any which way, air traffic control routes planes around the US on “highways in the sky”).

A typical problem, then, is to route multiple robots around a fixed set of paths on a factory floor so that they all get place to place without running into each other. To a mathematician, this is a problem about moving around on a graph. Each vertex is a destination, and each edge from one vertex to another is a path from one place to another. So it might look like this (the numbers on each edge is the length of that edge):

We can then think of robots in this picture as little disks moving along these paths. Of course, the whole problem is that you don’t want the robots running in to each other. This means each robot’s disk should bigger or smaller as needed to represent the size of the robot. A graph with distances is called a metric graph and moving robots of various sizes around on such a graph is sometimes called “Pebble Motion on Graphs Problems”.

Actually developing a routing algorithm for moving the robots around is a very challenging problem. It is already interesting and difficult to understand which routings are *even* *possible*.

How do we tackle such a problem? Mathematicians (like Drs. Dover and Özaydın) do something bold which actually simplifies the problem (once you get your mind around it). They consider the set of *all* possible configurations of robot positions on the graph.

On the one hand, this a a much bigger and more complicated gadget, but on the other hand it is topological space called a configuration space. This means you can unleash all the tools we have from topology to study the possible configurations of robots.

For example, a path in the configuration space translates into the simultaneous motion of the robots around the graph. So knowing that there is a path between any two configurations in the configuration space tells you that it is possible to move the robots into any positions you like.

The first question, then, is how many different configuration spaces are there for a fixed graph if you vary the size of the robots. To go back to our example of the factory, if you you replace your robots with new and improved models which are half the size of the old ones, does your old software still work, or are their new paths and configurations possible? For example, if you have a single track of length 1 meter:

If you have a two gigantic robots which are each 3/4 meter across, then they can’t both fit on this track. So there is *zero* possible configuration spaces for two big robots on this graph. But if you replace them with two small robots (say of size 1 cm), then they now both fit on the graph and so you have *at least one* space of possible configurations.

As a topological space, you count spaces the same if they are homeomorphic. So the first problem to ask (and which Drs. Dover and Özaydın answer) is how many non-homeomorphic configuration spaces are there?

You might think that there is infinitely many. Or at least many, many, many. In his 2011 PhD thesis at the University of Durham, Ken Deeley proved* that if you consider the first interesting case of two robots (one robot is easy!), then the number of possible configurations is actually finite.

Dr. Deeley gave an upper bound which is an exponential function in the number of edges on the graph. The good news is that it only depends on the number of edges, not how they are connected or how long they are or anything else. The bad news — as we all know — is that exponential functions get very big very fast. So if it really is an exponential number of configuration spaces, then it is surely impractical to handle them on a case by case base.

Drs. Dover and Özaydın took on the challenge. They prove the following amazing result:

**Theorem:** If you have n robots traveling around a graph, then the number of possible configuration spaces is bounded by a *polynomial of degree n* in the number of edges in the graph.

For example, in Deeley’s case of two robots, the bound is a quadratic polynomial! In fact, they even compute the bound for two robots on a graph with E edges in their paper and get that the number of configuration spaces is no more than:

That’s a heck of a lot less than an exponential bound!

A couple of notes before we go:

- They give an example with two robots where the number of configurations is bounded
*below* by a quadratic polynomial. This shows that you can’t get it down to a linear function.

- They also consider the case when the robots have various sizes. In fact, they even allow you to set how close each pair of robots can get to each other. Let’s say robots A, B, and C are all small, but A is carrying radioactive material. If robot B is sensitive and robot C is not sensitive to radioactivity, you can set the distance between A and B to be larger than the distance between A and C and B and C. Their theorem still applies!

Very nice result!

“Imagine if all these fish were mathematical theorems…” — James Dover

* To be 100% mathematically honest, Deeley considered spaces up to homotopy, not homeomorphism. There are fewer homotopy types, so a bound on homotopy types is not quite as strong a result as on homeomorphism types. That is, Drs. Dover and Özaydın result implies Dr. Deeley’s but not vice versa.