Minimum absolute value of subset sums of integer values
We give a Turing reduction from the $\mathrm{SubsetSum}$ problem.
Suppose we are given a $\mathrm{SubsetSum}$ instance $(A, k)$ where w.l.o.g. $A$ only contains positive integers, i.e. we want to find a set $X \subseteq A$ such that $\sum_{x \in X} x = k$ and define the set $A' = A \cup \{- k\}$.
We want to show that $f(A') = 0$ if and only if $(A, k) \in \mathrm{SubsetSum}$:

Suppose that $(A, k) \in \mathrm{SubsetSum}$. Then there exists a set $X \subseteq A$ such that $\sum_{x \in X} x = k$, i.e. the sum over the elements of $X \cup \{-k\} \subseteq A'$ is $0$, implying $f(A') = 0$.
Otherwise, we have $(A, k) \notin \mathrm{SubsetSum}$. As $A$ only contains positive integers but no subset of $A$ sums to $k$ we infer that any subset of $A'$ sums to some nonzero value.

It follows that if $f$ is computable by some polynomial time DTM then $\mathrm{SubsetSum}$ can in turn be decided in polynomial time, showing that $\mathsf P = \mathsf{NP}$.