I have a problem that can be solved with linear programming, but I'm hoping there's a combinatorial algorithm for this (even approximation is fine).
This is basically a load balancing problem using concepts from the set cover problem. Let $U$ be the set of elements and $S$ be the set of sets whose union is $U$ (just like in set cover). Let $x_i$ be a weight associated with $s_i \in S$, where $i \ge 1$, so $x_0$ is not an actual weight. Then the LP I am solving is as follows:
$$ \min x_0 \\ \sum_{i:u \in s_i}x_i \le x_0 \quad \forall u \in U \\ \sum_{i \ge 1} x_i = 1 \\ x_i \ge 0 $$
Intuitively, I am trying to "weight" my sets such that no single element gets overloaded. I am minimizing the maximum weight of some element. The weight of an element is the sum of the weights of the sets that contain that element.