Envy free chain store pricing

In this paper we study envy free pricing problem in general graphs where there is not a seller in every graph´s nodes. We assume unique establishment cost for initiating a store in each node and we wish to find an optimal set of nodes in which we would make the maximum profit by initiating stores in them. Our model is motivated from the observation that a same product has different prices in different locations and there is also an establishing cost for initiating any store. We consider both of these issues in our model: first where should we establish the stores, and second at what price should we sell our items in them to gain maximum possible profit. We prove that in a case of constant price our problem is NP-Hard and we present a (1 - 1/e)-approximation algorithm for solving “Equal prices-Equal costs” and “Equal prices-Difference costs” versions of this problem.