$\{\pm 1\}$-weighted zero-sum constants

Let $A,B\subseteq {\mathbb{Z}}_n\setminus \{0\}$. A sequence $S=(x_1,\dots ,x_k)$ in ${\mathbb{Z}}_n$ is called an $(A,B)$-weighted zero-sum sequence if there exist $a_1,\dots ,a_k\in A$ and $b_1,\dots ,b_k\in B$ such that $a_1{x}_1+\cdots +a_k{x}_k=0$ and $b_1{a}_1+\cdots +b_k{a}_k=0$. The constant $E_{A,B}(n)$ is defined to be the smallest positive integer $k$ such that every sequence of length $k$ in ${\mathbb{Z}}_n$ has an $(A,B)$-weighted zero-sum subsequence of length $n$. We determine the constant $E_{A,B}(n)$ and the related constants $C_{A,B}(n)$ and $D_{A,B}(n)$ when $A=\{\pm 1\}$ and $B=\{1\}$.