Travelling Salesman Problem
The Traveling Salesman Problem is a well known challenge in Computer Science: it consists on finding the shortest route possible that traverses all cities in a given map only once. Although its simple explanation, this problem is, indeed, NP-Complete. This implies that the difficulty to solve it increases rapidly with the number of cities, and we do not know in fact a general solution that solves the problem. For that reason, we currently consider that any method able to find a sub-optimal solution is generally good enough (we cannot verify if the solution returned is the optimal one most of the times).
To solve it, we can try to apply a modification of the Self-Organizing Map (SOM) technique.
Arora PTAS for Euclidean TSP
The Travelling Salesman Problem (TSP) is one of the most famous problems in Computer Science, but it turns out to be NP-Hard. It's even NP-hard to approximate it to any polynomial factor in the general case! Thankfully we can do a constant (around 1.5) approximation when the distances for which we are solving the problem come from a metric. While exactly what this constant is remains open, we know we cannot have a PTAS in the general case. What's a PTAS? It's a Polynomial Time Approximation Scheme. The idea is that you give me an epsilon, I will give you a (1+epsilon) approximation algorithm whose runtime depends on epsilon but is polynomial in n. So we have runtimes like poly(n) 2^{1/epsilon} and others.
Sanjeev Arora discovered a PTAS for TSP when the distances come from a Euclidean space a couple of decades ago. This is very good news for Uber and the like, since their distances usually come from the plane! The idea is not hard to understand, and I plan to make the talk accessible to anyone who is comfortable with Dynamic Programming.