Approximating the Revenue Maximization Problem with Sharp Demands

We consider the revenue maximization problem with sharp multi-demand, in which $m$ indivisible items have to be sold to $n$ potential buyers. Each buyer $i$ is interested in getting exactly $d_i$ items, and each item $j$ gives a benefit $v_{ij}$ to buyer $i$. We distinguish between unrelated and related valuations. In the former case, the benefit $v_{ij}$ is completely arbitrary, while, in the latter, each item $j$ has a quality $q_j$, each buyer $i$ has a value $v_i$ and the benefit $v_{ij}$ is defined as the product $v_i q_j$. The problem asks to determine a price for each item and an allocation of bundles of items to buyers with the aim of maximizing the total revenue, that is, the sum of the prices of all the sold items. The allocation must be envy-free, that is, each buyer must be happy with her assigned bundle and cannot improve her utility. We first prove that, for related valuations, the problem cannot be approximated to a factor $O(m^{1-\epsilon})$, for any $\epsilon>0$, unless {\sf P} = {\sf NP} and that such result is asymptotically tight. In fact we provide a simple $m$-approximation algorithm even for unrelated valuations. We then focus on an interesting subclass of "proper" instances, that do not contain buyers a priori known not being able to receive any item. For such instances, we design an interesting $2$-approximation algorithm and show that no $(2-\epsilon)$-approximation is possible for any $0<\epsilon\leq 1$, unless {\sf P} $=$ {\sf NP}. We observe that it is possible to efficiently check if an instance is proper, and if discarding useless buyers is allowed, an instance can be made proper in polynomial time, without worsening the value of its optimal solution.