Some optimal inapproximability results
Journal of the ACM (JACM)
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SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Combination can be hard: approximability of the unique coverage problem
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Single-minded unlimited supply pricing on sparse instances
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
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STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Maximum directed cuts in acyclic digraphs
Journal of Graph Theory
Beating the Random Ordering is Hard: Inapproximability of Maximum Acyclic Subgraph
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
A quasi-PTAS for profit-maximizing pricing on line graphs
ESA'07 Proceedings of the 15th annual European conference on Algorithms
Pricing commodities, or how to sell when buyers have restricted valuations
WAOA'07 Proceedings of the 5th international conference on Approximation and online algorithms
Near-optimal pricing in near-linear time
WADS'05 Proceedings of the 9th international conference on Algorithms and Data Structures
On Profit-Maximizing Pricing for the Highway and Tollbooth Problems
SAGT '09 Proceedings of the 2nd International Symposium on Algorithmic Game Theory
A sublogarithmic approximation for highway and tollbooth pricing
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
On the hardness of pricing loss-leaders
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
On the complexity of the highway problem
Theoretical Computer Science
A path-decomposition theorem with applications to pricing and covering on trees
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
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We consider the following item pricing problem which has received much attention recently. A seller has an infinite numbers of copies of n items. There are m buyers, each with a budget and an intention to buy a fixed subset of items. Given prices on the items, each buyer buys his subset of items, at the given prices, provided the total price of the subset is at most his budget. The objective of the seller is to determine the prices such that her total profit is maximized. In this paper, we focus on the case where the buyers are interested in subsets of size at most two. This special case is known to be APX-hard (Guruswami et al [1]). The best known approximation algorithm, by Balcan and Blum, gives a 4-approximation [2]. We show that there is indeed a gap of 4 for the combinatorial upper bound used in their analysis. We further show that a natural linear programming relaxation of this problem has an integrality gap of 4, even in this special case. Then we prove that the problem is NP-hard to approximate within a factor of 2 assuming the Unique Games Conjecture; and it is unconditionally NP-hard to approximate within a factor 17/16. Finally, we extend the APX-hardness of the problem to the special case in which the graph formed by items as vertices and buyers as edges is bipartite . We hope that our techniques will be helpful for obtaining stronger hardness of approximation bounds for this problem.