On the complexity of approximating the independent set problem
Information and Computation
Randomized algorithms
Computational Complexity
Combinatorial optimization
Proof verification and the hardness of approximation problems
Journal of the ACM (JACM)
Algorithmic mechanism design (extended abstract)
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Competitive auctions and digital goods
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Relations between average case complexity and approximation complexity
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Approximating the Domatic Number
SIAM Journal on Computing
Competitive Auctions for Multiple Digital Goods
ESA '01 Proceedings of the 9th Annual European Symposium on Algorithms
Algorithms for approximate graph coloring
Algorithms for approximate graph coloring
Ruling Out PTAS for Graph Min-Bisection, Densest Subgraph and Bipartite Clique
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
On profit-maximizing envy-free pricing
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Mechanism Design via Machine Learning
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Single-minded unlimited supply pricing on sparse instances
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
A Nonparametric Approach to Multiproduct Pricing
Operations Research
Algorithmic pricing via virtual valuations
Proceedings of the 8th ACM conference on Electronic commerce
Item pricing for revenue maximization
Proceedings of the 9th ACM conference on Electronic commerce
Optimal envy-free pricing with metric substitutability
Proceedings of the 9th ACM conference on Electronic commerce
Combination Can Be Hard: Approximability of the Unique Coverage Problem
SIAM Journal on Computing
On Profit-Maximizing Pricing for the Highway and Tollbooth Problems
SAGT '09 Proceedings of the 2nd International Symposium on Algorithmic Game Theory
The Stackelberg Minimum Spanning Tree Game
Algorithmica
Pricing on paths: a PTAS for the highway problem
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Near-optimal pricing in near-linear time
WADS'05 Proceedings of the 9th international conference on Algorithms and Data Structures
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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 $\varepsilon0$, 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 $\mathrm{NP}\nsubseteq\bigcap_{\delta0}\mathrm{BPTIME}(2^{\mathcal{O}(n^{\delta})})$. 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.