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.

A while ago, someone in a Discord server I’m in asked how much of the math behind a system you need to know to implement it. I thought it was an interesting question, and I felt qualified to answer it, so I ended up writing quite a lengthy response. It just occurred to me that it might also be useful to other people, so I thought I would clean it up a little bit and archive it here.

While all NP-hard optimization problems are identical in terms of exact solvability, they may differ wildly from the approximative point of view. If the goal is to obtain an answer that is “good enough”, some problems become much easier (such as KNAPSACK), while others (such as CLIQUE) remain extremely hard.