Buying Cheap Is Expensive: Approximability of Combinatorial Pricing Problems

We investigate nonparametric multiproduct pricing problems, in which we want to find revenue maximizing prices for products $\mathcal{P}$ based on a set of customer samples $\mathcal{C}$. We mostly focus on the unit-demand case, in which products constitute strict substitutes and each customer aims to purchase a single product. In this setting a customer sample consists of a number of nonzero values for different products and possibly an additional product ranking. Once prices are fixed, each customer chooses to buy one of the products she can afford based on some predefined selection rule. We distinguish between the min-buying, max-buying, and rank-buying models. Some of our results also extend to single-minded pricing, in which case products are strict complements and every customer seeks to buy a single set of products, which she purchases if the sum of prices is below her valuation for that set. For the min-buying model we show that the revenue maximization problem is not approximable within factor $\mathcal{O}(\log^{\varepsilon}|\mathcal{C}|)$ for some constant $\varepsilon>0$, unless $\mathrm{NP}\subseteq\mathrm{DTIME}(n^{\mathcal{O}(\log\log n)})$, thereby almost closing the gap between the known algorithmic results and previous lower bounds. We also prove inapproximability within $\mathcal{O}(\ell^{\varepsilon})$, $\ell$ being an upper bound on the number of nonzero values per customer, and $\mathcal{O}(|\mathcal{P}|^{\varepsilon})$ under slightly stronger assumptions and provide matching upper bounds. Surprisingly, these hardness results hold even if a price ladder constraint, i.e., a predefined order on the prices of all products, is given. Without the price ladder constraint we obtain similar hardness results for the special case of uniform valuations, i.e., the case that every customer has identical values for all the products she is interested in, assuming specific hardness of the balanced bipartite independent set problem in constant degree graphs or hardness of refuting random 3CNF formulas. Introducing a slightly more general problem definition in which customers are given as an explicit probability distribution, we obtain inapproximability within $\mathcal{O}(|\mathcal{P}|^{\varepsilon})$ assuming $\mathrm{NP}\nsubseteq\bigcap_{\delta>0}\mathrm{BPTIME}(2^{\mathcal{O}(n^{\delta})})$. These results apply to single-minded pricing as well. For the max-buying model a polynomial-time approximation scheme exists if a price ladder is given. We give a matching lower bound by proving strong NP-hardness. Assuming limited product supply, we analyze a generic local search algorithm and prove that it is 2-approximate. Finally, we discuss implications for the rank-buying model.