In Part 3, we discussed the densest subgraph problem (DSP) and some algorithms for solving it. In this section, we’ll be looking at how we can generalize this problem and those algorithms to solve some other problems. This is the part of the talk where I will introduce some fancy new math I had to learn in order to understand how one might generalize iterative peeling to solve adjacent problems.

I keep coming across articles (and occasionally videos, and occasionally fiction or poetry) online that I want to share and comment on, but I recently realized that I will never have time to fully comment on everything I read that I find interesting. So I’m pulling another page out of Cory Doctorow’s book: here is a dump of links to cool stuff, along with some (hopefully very) brief descriptions of why I found these articles interesting. (Also, here is a link to a link dump where Doctorow ¹ talks about link blogging. It might be interesting for you to look at.)

In Part 2 of this talk, we gave a crash course to graph theory and showed how we can use it to view some real-world problems as instances of the densest subgraph problem (DSP). But what exactly is the DSP? If you’ve studied graph theory, you may have heard of something called the Maximum Clique Problem. The goal of the max clique problem is to find the largest complete subgraph in a graph. If we consider our vertices to be people, and edges to represent a friendship relationship between two people, in the max clique problem we are trying to find the largest friend group in a community.

In the first part of this talk, I made the case that theory is useful because it allows us to find (or at the very least, have the correct toolkit and language to explore) solutions to real-world problems. In this part, we are going to look at some examples of such problems and develop mathematical language to be able to discuss them more abstractly. I’ve put the term “real-world” in quotes in the title, because I’m going to be talking about these problems in a lot of generality. However, I want to stress that specific instances of these problems are actually relevant in industry, and I think it’ll be easy to see why once I start talking about them.

If you’re a computer science student, you probably had to take an introductory discrete math course at some point. Did you enjoy it? If so, this talk probably isn’t for you, so you can feel free to skip the rest. (Or not – hopefully you feel like you can still learn something from me!) Jokes aside, it’s actually okay not to enjoy your intro to discrete math course: like, personally, I loved mine, but I also completely hated my discrete probability course and would prefer never to see it again. But I pick on discrete math because I feel like if it’s taught well, it can be a turning point for many people, and it certainly was for me.