Here is a simple-to-state, but notoriously difficult open question from algorithmics, which I think might not have received yet the attention of the larger mathematics community.

The Greedy Superstring Conjecture:

Consider a set of $n$ strings $s_1, \dots, s_n$ over a finite alphabet $\Sigma$. Assume that none of the strings is a substring of another. We want to find a shortest string $s$ that contains all $s_1, \dots, s_n$ as substrings.

Here's an "obvious" heuristic: find the two strings $s_i, s_j$ with the largest overlap, and replace $s_i$ and $s_j$ with the string obtained by overlapping $s_i$ and $s_j$ as much as possible. After $n-1$ steps we obtain a string that is the superstring of all $s_1, \dots, s_n$. The conjecture says that this solution is at most twice as long as the optimum.

The problem is described (among other places) in Vazirani's book Approximation Algorithms, Chapter 2.3.