On a Communication Complexity problem in Combinatorial Number Theory

The original knapsack problem is well known to be NP-complete. In a multidimensional version one have to decide whether a $p\in {\mathbb{N}}^k$ is in a sumset-sum of a set $X\subseteq {\mathbb{N}}^k$ or not. In this paper we are going to investigate a communication complexity problem related to this.