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17
A sublogarithmic approximation for highway and tollbooth pricing
 In Proceedings of the 37th International Colloquium on Automata, Languages and Programming
, 2010
"... An instance of the tollbooth problem consists of an undirected network and a collection of singleminded customers, each of which is interested in purchasing a fixed path subject to an individual budget constraint. The objective is to assign a perunit price to each edge in a way that maximizes the c ..."
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An instance of the tollbooth problem consists of an undirected network and a collection of singleminded customers, each of which is interested in purchasing a fixed path subject to an individual budget constraint. The objective is to assign a perunit price to each edge in a way that maximizes the collective revenue obtained from all customers. The revenue generated by any customer is equal to the overall price of the edges in her desired path, when this cost falls within her budget; otherwise, that customer will not purchase any edge. Our main result is a deterministic algorithm for the tollbooth problem on trees whose approximation ratio is O(log m / log log m), where m denotes the number of edges in the underlying graph. This finding improves on the currently best performance guarantees for trees, due to Elbassioni et al. (SAGT ’09), as well as for paths (commonly known as the highway problem), due to Balcan and Blum (EC ’06). An additional interesting consequence is a computational separation between tollbooth pricing on trees and the original prototype problem of singleminded unlimited supply pricing, under a plausible hardness hypothesis due to Demaine et al. (SODA ’06).
EnvyFree Allocations for Budgeted Bidders
"... We study the problem of identifying prices to support a given allocation of items to bidders in an envyfree way. A bidder will envy another bidder if she would prefer to obtain the other bidder’s item at the price paid by that bidder. Envyfree prices for allocations have been studied extensively; ..."
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We study the problem of identifying prices to support a given allocation of items to bidders in an envyfree way. A bidder will envy another bidder if she would prefer to obtain the other bidder’s item at the price paid by that bidder. Envyfree prices for allocations have been studied extensively; here, we focus on the impact of budgets: beyond their willingness to pay for items, bidders are also constrained by their ability to pay, which may be lower than their willingness. In a recent paper, Aggarwal et al. show that a variant of the Ascending Auction finds a feasible and bidderoptimal assignment and supporting envyfree prices in polynomial time so long as the input satisfies certain nondegeneracy conditions. While this settles the problem of finding a feasible allocation, an auctioneer might sometimes also be interested in a specific allocation of items to bidders. In this paper, we therefore study the problem of whether a given allocation can be supported with envyfree prices. We present two polynomialtime algorithms for this problem, one which finds maximal prices supporting the given allocation (if such prices exist), and another which finds minimal prices. We also prove a structural result characterizing when different allocations are supported by the same minimal price vector. 1
EnvyFree Pricing in Multiitem Markets
, 2010
"... In this paper, we study revenue maximizing envyfree pricing in multiitem markets: There are m indivisible items and n potential buyers where each buyer is interested in acquiring one item. The goal is to determine allocations (a matching between buyers and items) and prices of all items to maximiz ..."
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In this paper, we study revenue maximizing envyfree pricing in multiitem markets: There are m indivisible items and n potential buyers where each buyer is interested in acquiring one item. The goal is to determine allocations (a matching between buyers and items) and prices of all items to maximize total revenue given that all buyers are envyfree. We give a polynomial time algorithm to compute a revenue maximizing envyfree pricing when every buyer evaluates at most two items at a positive valuation, by reducing it to an instance of weighted independent set in a perfect graph and applying the Strong Perfect Graph Theorem. We complement our result by showing that the problem becomes NPhard if some buyers are interested in at least three items. Next we extend the model to allow buyers to demand a subset of consecutive items, motivated from TV advertising where advertisers are interested in different consecutive slots with different valuations and lengths. We show that the revenue maximizing envyfree pricing in this setting can be computed in polynomial time.
Improved Hardness Results for Profit Maximization Pricing Problems with Unlimited Supply
"... 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 Sc ⊆ [n] of items of interest, together with a budget Bc, and we assume that there is an unlimited supply of each item. Once the prices are fixe ..."
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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 Sc ⊆ [n] of items of interest, together with a budget Bc, 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 Sc, according to its buying rule. The goal is to set the item prices so as to maximize the total profit. We study the unitdemand minbuying pricing (UDPMIN) and the singleminded pricing (SMP) problems. In the former problem, each customer c buys the cheapest item i ∈ Sc, if its price is no higher than the budget Bc, and buys nothing otherwise. In the latter problem, each customer c buys the whole set Sc if its total price is at most Bc, 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−ɛ (m+n) hard to approximate for any constant ɛ, unless NP ⊆ DTIME(n logδ n where δ is a constant depending on ɛ. Restricting our attention to approximation factors depending only on n, we show that these problems are 2log1−δ nhard to approximate for any δ> 0 unless NP ⊆ ZPTIME(nlogδ ′ n ′), where δ is some constant depending on δ. We also prove that
On Revenue Maximization with Sharp MultiUnit Demands
 In CoRR, arXiv:arXiv:1210.0203
, 2012
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Envy, Multi Envy, and Revenue Maximization
, 2009
"... We study the envy free pricing problem faced by a seller who wishes to maximize revenue by setting prices for bundles of items. Consistent with standard usage [9] [10], we define an allocation/pricing to be envy free if no agent wishes to replace her allocation (and pricing) with those of another a ..."
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We study the envy free pricing problem faced by a seller who wishes to maximize revenue by setting prices for bundles of items. Consistent with standard usage [9] [10], we define an allocation/pricing to be envy free if no agent wishes to replace her allocation (and pricing) with those of another agent. If there is an unlimited supply of items and agents are single minded then we show that finding the revenue maximizing envy free allocation/pricing can be solved in polynomial time by reducing it to an instance of weighted independent set on a perfect graph. We also consider a generalization of envy freeness. We define an allocation/pricing as multi envy free if no agent wishes to replace her allocation with the union of the allocations of some set of other agents and her price with the sum of their prices. We show that even though such allocation/pricing can be approximated by O(log m + log n) factor [3], it is coNPhard to decide if a given allocation/pricing is multi envy free. We also show that revenue maximization multi envy free allocation/pricing is APX hard. An interesting restricted version of the subset pricing problem is when all items are intervals of a line segment and all requests are a contiguous set of items along the line. The motivation here is that one can think of the agents as drivers on a highway when each product is highway segment(or guests in a hotel — items are translated to dates). In this setting, determining if a given allocation/pricing is multi envy free is polynomial time. If the highway has bounded capacities then a revenue maximizing envy free allocation/pricing can be computed in polynomial time and we also give an FPTAS for the revenue maximizing multi envy free allocation/pricing.
Envyfree Pricing with General Supply Constraints for Unit Demand Consumers
 JOURNAL OF COMPOUTER SCIENCE AND TECHNOLOGY
, 2011
"... The envyfree pricing problem can be stated as finding a pricing and allocation scheme in which each consumer is allocated a set of items that maximize her utility under the pricing. The goal is to maximize seller revenue. We study the problem with general supply constraints which are given as an in ..."
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The envyfree pricing problem can be stated as finding a pricing and allocation scheme in which each consumer is allocated a set of items that maximize her utility under the pricing. The goal is to maximize seller revenue. We study the problem with general supply constraints which are given as an independence system defined over the items. (See Definition 2.) The constraints, for example, can be a number of linear constraints or matroids. This captures the situation where items do not preexist, but are produced in reflection of consumer valuation of the items under the limit of resources. This paper focuses on the case of unitdemand consumers. In the setting, there are n consumers and m items; each item may be produced in multiple copies. Each consumer i ∈ [n] has a valuation vij on item j in the set Si in which she is interested. She must be allocated (if any) an item which gives the maximum (nonnegative) utility. Suppose we are given an αapproximation oracle for finding the maximum weight independent set for the given independence system (or a slightly stronger oracle); for a large number of natural and interesting supply constraints, constant approximation algorithms are available. We obtain the following results. • O(α log n)approximation for the general case. • O(αk)approximation when each consumer is interested in at most k distinct types of items. • O(αf)approximation when each type of item is interesting to at most f consumers. Note that the final two results were previously unknown even without the independence system constraint.
EnvyFree Makespan Approximation
, 2009
"... We study envyfree mechanisms for scheduling tasks on unrelated machines (agents) that approximately minimize the makespan. For indivisible tasks, we put forward an envyfree polytime mechanism that approximates the minimal makespan to within a factor of O(log m), where m is the number of machines. ..."
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We study envyfree mechanisms for scheduling tasks on unrelated machines (agents) that approximately minimize the makespan. For indivisible tasks, we put forward an envyfree polytime mechanism that approximates the minimal makespan to within a factor of O(log m), where m is the number of machines. We also show a lower bound of Ω(log m / log log m). This improves the recent result of Hartline et al. [15] who give an upper bound of (m+1)/2, and a lower bound of 2−1/m. For divisible tasks, we show that there always exists an envyfree polytime mechanism with optimal makespan.
Competitive Algorithms for Online Pricing
"... Abstract. Given a seller with m amount of items, a sequence of users {u1,u2,...} come one by one, the seller must set the unit price and assign some amount of items to each user on his/her arrival. Items can be sold fractionally. Each ui has his/her value function vi(·) such that vi(x) is the highes ..."
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Abstract. Given a seller with m amount of items, a sequence of users {u1,u2,...} come one by one, the seller must set the unit price and assign some amount of items to each user on his/her arrival. Items can be sold fractionally. Each ui has his/her value function vi(·) such that vi(x) is the highest unit price ui is willing to pay for x items. The objective is to maximize the revenue by setting the price and amount of items for each user. In this paper, we have the following contributions: if the highest value h among all vi(x) is known in advance, we first show the lower bound of the competitive ratio is O(log h), then give an online algorithm with competitive ratio O(log h); if h is not known in advance, we give an online algorithm with competitive ratio O(h 3log−1/2 h 1