Vector Subset Sum Decision Problem

Definition

Vector Subset Sum Decision Problem

Instance: A dimension $d\ge 1$, an indexed family of vectors $p_1,\dots ,p_r\in {\mathbb{Z}}^d$, and a target $\tau \in {\mathbb{Z}}^d$, with integers encoded in binary unless specified otherwise. The dimension is part of the input, not a fixed constant.

Question: Does there exist a set of indices $I\subseteq \{1,\dots ,r\}$ such that

$$\sum _{i\in I}{p}_i=\tau ?$$

Equality must hold in every coordinate. Each index may be selected at most once; equal vectors at distinct indices remain independently selectable. This is the vector-valued version of the subset sum decision problem.