An approximate cost recovery scheme for the k-product facility location game with penalties

This paper considers a variant of the classic facility location game called the k-product facility location game with penalties. In this game, each facility can only produce at most one kind of product and each customer must be supplied with k different products or be rejected by paying the penalty cost. The authors propose a 2-approximate cross-monotonic cost-sharing scheme for the game when $k=2$, and a ($\frac{3}{2}k-\frac{3}{2}$)-approximate cross-monotonic cost-sharing scheme when $k\ge 3$, respectively. Furthermore, the lower bound is obtained for the k-product facility location game with penalties.