Suppose, i have a combinatoric problem: given a set of $n$ tuples $Z =\{(a_1, b_1), \cdots, (a_n, b_n)\}, a_i \in \mathbb{R}^+, b_i \in \mathbb{R}^+$.
My goal is to find a $S \in \mathcal{P}(Z)$, such that $\sum_{i \in S} a_i + \sum_{i \in S^c} b_i $ is minimized. Here $\mathcal{P}(.)$ is powerset and $S^c$ is the complement of $S$ over $Z$.
Since polynomial is a very nice tool to encode combinatoric problems, wondering if there are any formulation does this.
I looked a little into tropical geometry. it seems I can easily achieve $\sum_{i \in S} a_i$ using the tropical polynomial. But not sure how to include the complement as well.