Algorithms for Multi-product Pricing

Aggarwal, Gagan; Feder, Tomás; Motwani, Rajeev; An, Zhijie

doi:10.1007/978-3-540-27836-8_9

Cited by 70 publications

(126 citation statements)

References 6 publications

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“…We refer to the corresponding pricing problem as UDP MIN . This problem was first introduced by Rusmevichientong et al [22,23], and subsequently Aggarwal et al [1] have shown an O(log m + log n)-approximation algorithm for it.…”

Section: Introductionmentioning

confidence: 99%

“…Here, each customer c buys the whole set S c of items if its total price does not exceed its budget B c , and buys nothing otherwise. This problem was introduced by Guruswami et al [17], who also show that the techniques of [1] can be used to obtain an O(log m + log n)-approximation algorithm for SMP. Hartline and Koltun [18] gave a (1 + )-approximation algorithm for both UDP MIN and SMP when the number of items n is constant.…”

Section: Introductionmentioning

confidence: 99%

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Improved Hardness Results for Profit Maximization Pricing Problems with Unlimited Supply

Chalermsook

Chuzhoy

Kannan

et al. 2012

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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We consider profit maximization pricing problems, where we are given a set of m customers and a set of n items. Each customer c is associated with a subset S c ⊆ [n] of items of interest, together with a budget B c , and we assume that there is an unlimited supply of each item. Once the prices are fixed for all items, each customer c buys a subset of items in S c , according to its buying rule. The goal is to set the item prices so as to maximize the total profit.We study the unit-demand min-buying pricing (UDP MIN ) and the single-minded pricing (SMP) problems. In the former problem, each customer c buys the cheapest item i ∈ S c , if its price is no higher than the budget B c , and buys nothing otherwise. In the latter problem, each customer c buys the whole set S c if its total price is at most B c , and buys nothing otherwise. Both problems are known to admit O(min {log(m + n), n})-approximation algorithms. We prove that they are log 1− (m+n) hard to approximate for any constant , unless NP ⊆ DTIME(n log δ n ), where δ is a constant depending on . Restricting our attention to approximation factors depending only on n, we show that these problems are 2 log 1−δ n -hard to approximate for any δ > 0 unless NP ⊆ ZPTIME(n log δ n ), where δ is some constant depending on δ. We also prove that restricted versions of UDP MIN and SMP, where the sizes of the sets S c are bounded by k, are k 1/2− -hard to approximate for any constant . We then turn to the Tollbooth Pricing problem, a special case of SMP, where each item corresponds to an edge in the input graph, and each set S c is a simple path in the graph. We show that Tollbooth Pricing is at least as hard to approximate as the Unique Coverage problem, thus obtaining an Ω(log n)-hardness of approximation, assuming NP ⊆ BPTIME(2 n δ ), for any constant δ, and some constant depending on δ.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Improved Hardness Results for Profit Maximization Pricing Problems with Unlimited Supply

Chalermsook

Chuzhoy

Kannan

et al. 2012

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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“…The notion of algorithmic pricing encompasses a wide range of optimization problems aiming to assign revenue-maximizing prices to some fixed set of items given information about the valuation functions of potential customers [1,13]. In a line of recent work the approximation complexity of this kind of problem has received considerable attention.…”

Section: Introductionmentioning

confidence: 99%

Improved Hardness of Approximation for Stackelberg Shortest-Path Pricing

Briest

Chalermsook

Khanna

et al. 2010

Lecture Notes in Computer Science

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We consider the Stackelberg shortest-path pricing problem, which is defined as follows. Given a graph G with fixed-cost and pricable edges and two distinct vertices s and t, we may assign prices to the pricable edges. Based on the predefined fixed costs and our prices, a customer purchases a cheapest s-t-path in G and we receive payment equal to the sum of prices of pricable edges belonging to the path. Our goal is to find prices maximizing the payment received from the customer. While Stackelberg shortest-path pricing was known to be APX-hard before, we provide the first explicit approximation threshold and prove hardness of approximation within 2−o(1). We also argue that the nicely structured type of instance resulting from our reduction captures most of the challenges we face in dealing with the problem in general and, in particular, we show that the gap between the revenue of an optimal pricing and the only known general upper bound can still be logarithmically large. Disciplines Computer SciencesComments Briest, P., CHalermsook, P., Khanna, S., Laekhanukit, B., & Nanongkai, D., Improved Abstract. We consider the Stackelberg shortest-path pricing problem, which is defined as follows. Given a graph G with fixed-cost and pricable edges and two distinct vertices s and t, we may assign prices to the pricable edges. Based on the predefined fixed costs and our prices, a customer purchases a cheapest s-t-path in G and we receive payment equal to the sum of prices of pricable edges belonging to the path. Our goal is to find prices maximizing the payment received from the customer. While Stackelberg shortest-path pricing was known to be APX-hard before, we provide the first explicit approximation threshold and prove hardness of approximation within 2 − o(1). We also argue that the nicely structured type of instance resulting from our reduction captures most of the challenges we face in dealing with the problem in general and, in particular, we show that the gap between the revenue of an optimal pricing and the only known general upper bound can still be logarithmically large.

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“…To provide a reader more insight on item-pricing problems and related Stackelberg games we refer to the introductions in [3,4,9]. For surveys on the solution methods developed for the geometric k-median and facility location problems we refer to [1,2,6,8].…”

Section: Introductionmentioning

confidence: 99%

Location, pricing and the problem of Apollonius

et al. 2017

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In Euclidean plane geometry, Apollonius' problem is to construct a circle in a plane that is tangent to three given circles. We will use a solution to this ancient problem to solve several versions of the following geometric optimization problem. Given is a set of customers located in the plane, each having a demand for a product and a budget. A customer is satisfied if her total, travel and purchase, costs do not exceed the budget. The task is to determine location of production facilities in the plane and one price for the product such that the revenue generated from the satisfied customers is maximized.Artem Panin: This work was partially supported by RFBR Grants 15-37-51018 mol_nr and 16-07-00319.

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Algorithms for Multi-product Pricing

Cited by 70 publications

References 6 publications

Improved Hardness Results for Profit Maximization Pricing Problems with Unlimited Supply

Improved Hardness Results for Profit Maximization Pricing Problems with Unlimited Supply

Improved Hardness of Approximation for Stackelberg Shortest-Path Pricing

Location, pricing and the problem of Apollonius

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