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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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On the Hardness of Pricing Lossleaders
, 2011
"... Consider the problem of pricing n items under an unlimited supply with m buyers. Each buyer is interested in a bundle of at most k of the items. These buyers are single minded, which means each of them has a budget and they will either buy all the items if the total price is within their budget or t ..."
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Cited by 3 (1 self)
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Consider the problem of pricing n items under an unlimited supply with m buyers. Each buyer is interested in a bundle of at most k of the items. These buyers are single minded, which means each of them has a budget and they will either buy all the items if the total price is within their budget or they will buy none of the items. The goal is to price each item with profit margin p1, p2,..., pn so as to maximize the overall profit. When k = 2, such a problem is called the graphvertexpricing problem. Another special case of the problem is the highwaypricing problem when the items (tollbooths) are arranged linearly on a line and each buyer (as a driver) is interested in paying for a path that consists of consecutive items. The goal again is to price the items (tolls) so as to maximize the total profits. There is an O(k)approximation algorithm by [BB06] when the price on each item must be above its margin cost; i.e., pi> 0 for every i ∈ [n]. As for the highway problem, a PTAS is shown in [GR11]. We investigate the above problem when the seller is allowed to price some of the items below their margin cost. It is shown in [BB06, BBCH07] that by pricing some of the items below cost, the maximum profit can increase by a factor of Ω(log n). These items sold
A PathDecomposition Theorem with Applications to Pricing and Covering on Trees
"... In this paper we focus on problems characterized by an input nnode tree and a collection of subpaths. Motivated by the fact that some of these problems admit a very good approximation (or even a polytime exact algorithm) when the input tree is a path, we develop a decomposition theorem of trees ..."
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Cited by 2 (2 self)
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In this paper we focus on problems characterized by an input nnode tree and a collection of subpaths. Motivated by the fact that some of these problems admit a very good approximation (or even a polytime exact algorithm) when the input tree is a path, we develop a decomposition theorem of trees into paths. Our decomposition allows us to partition the input problem into a collection of O(log logn) subproblems, where in each subproblem either the input tree is a path or there exists a hitting set F of edges such that each path has a nonempty, small intersection with F. When both kinds of subproblems admit constant approximations, our method implies an O(log log n) approximation for the original problem. We illustrate the above technique by considering two natural problems of the mentioned kind, namely Uniform Tree Tollbooth and Unique Tree Coverage. In Uniform Tree Tollbooth each subpath has a budget, where budgets are within a constant factor from each other, and we have to choose nonnegative edge prices so that we maximize the total price of subpaths whose budget is not exceeded. In Unique Tree Coverage each subpath has a weight, and the goal is to select a subset X of edges so that we maximize the total weight of subpaths containing exactly one edge of X. We obtain O(log logn) approximation algorithms for both problems. The previous best approximations are O(logn / log log n) by Gamzu and Segev [ICALP’10] and O(logn) by Demaine et al. [SICOMP’08] for the first and second problem, respectively, however both previous results were obtained for much more general problems with arbitrary budgets (weights).
An LProunding 2√2 Approximation for Restricted Maximum Acyclic Subgraph
, 2014
"... In the classical Maximum Acyclic Subgraph problem (MAS), given a directededge weighted graph, we are required to find an ordering of the nodes that maximizes the total weight of forwarddirected edges. MAS admits a 2approximation, and this approximation is optimal under the Unique Game Conjecture. ..."
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In the classical Maximum Acyclic Subgraph problem (MAS), given a directededge weighted graph, we are required to find an ordering of the nodes that maximizes the total weight of forwarddirected edges. MAS admits a 2approximation, and this approximation is optimal under the Unique Game Conjecture. In this paper we consider a generalization of MAS, theRestricted Maximum Acyclic Subgraph problem (RMAS), where each node is associated with a list of integer labels, and we have to find a labeling of the nodes so as to maximize the weight of edges whose head label is larger than the tail label. The interest in RMAS is mostly due to its connections with the Vertex Pricing problem (VP). VP is known to be (2 − )hard to approximate via a reduction from RMAS, and the best known approximation factor for both problems is 4 (which is achieved via fairly simple algorithms). In this paper we present a nontrivial LProunding algorithm for RMAS with approximation ratio 2 2 ≈ 2.828. Our result shows that, in order to prove a 4hardness of approximation result for VP (if possible), one should consider reductions from harder problems. Alternatively, our approach might suggest a different way to design approximation algorithms for VP.
Improved Approximation Algorithms for Unsplittable Flow on a Path with Time Windows
"... In the wellstudied Unsplittable Flow on a Path problem (UFP), we are given a path graph with edge capacities. Furthermore, we are given a collection of n tasks, each one characterized by a subpath, a weight, and a demand. Our goal is to select a maximum weight subset of tasks so that the total de ..."
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In the wellstudied Unsplittable Flow on a Path problem (UFP), we are given a path graph with edge capacities. Furthermore, we are given a collection of n tasks, each one characterized by a subpath, a weight, and a demand. Our goal is to select a maximum weight subset of tasks so that the total demand of selected tasks using each edge is upper bounded by the corresponding capacity. Chakaravarthy et al. [ESA’14] studied a generalization of UFP, bagUFP, where tasks are partitioned into bags, and we can select at most one task per bag. Intuitively, bags model jobs that can be executed at different times (with different duration, weight, and demand). They gave a O(logn) approximation for bagUFP. This is also the best known ratio in the case of uniform weights. In this paper we achieve the following main results: •We present an LPbased O(logn / log logn) approximation for bagUFP. We remark that, prior to our work, the best known integrality gap (for a nonextended formulation) was O(logn) even in the special case of UFP [Chekuri et al., APPROX’09]. •We present an LPbasedO(1) approximation for uniformweight bagUFP. This also generalizes the integrality gap bound for uniformweight UFP by Anagnostopoulos et al. [IPCO’13]. •We consider a relevant special case of bagUFP, twUFP, where tasks in a bag model the possible ways in which we can schedule a job with a given processing time within a given time window. We present a QPTAS for twUFP with quasipolynomial demands and under the Bounded TimeWindow Assumption, i.e. assuming that the time window size of each job is within a constant factor from its processing time. This generalizes the QPTAS for UFP by Bansal et al. [STOC’06].
Inapproximability Reductions and Integrality Gaps
, 2013
"... In this thesis we prove intractability results for several well studied problems in combinatorial optimization. Closest ..."
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In this thesis we prove intractability results for several well studied problems in combinatorial optimization. Closest
On the tollbooth problem
, 2012
"... We show that the gap in profit between a naturally defined upward instance of the tollbooth problem on a tree and the original instance can be at least Ω(log log n) where n is the number of edges in the tree. ..."
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We show that the gap in profit between a naturally defined upward instance of the tollbooth problem on a tree and the original instance can be at least Ω(log log n) where n is the number of edges in the tree.
Gap between upward and tollbooth instances
, 2011
"... We show that the gap in profit between a naturally defined upward instance of the tollbooth problem on a tree and the original instance can be at least Ω(log log n) where n is the number of edges in the tree. ..."
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We show that the gap in profit between a naturally defined upward instance of the tollbooth problem on a tree and the original instance can be at least Ω(log log n) where n is the number of edges in the tree.
Profitmaximizing pricing for tollbooths
"... The input to the tollbooth problem is a graph G = (V, E) and a set of m buyers Pi where each buyer is interested in buying a path Pi connecting si, ti ∈ V. Each buyer comes with a budget b(Pi), a positive real number. The problem is to set nonnegative prices to the edges E of G. A buyer buys her p ..."
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The input to the tollbooth problem is a graph G = (V, E) and a set of m buyers Pi where each buyer is interested in buying a path Pi connecting si, ti ∈ V. Each buyer comes with a budget b(Pi), a positive real number. The problem is to set nonnegative prices to the edges E of G. A buyer buys her path Pi if the sum of prices on Pi is at most her budget. If a buyer buys her path, we obtain a profit equal to the sum of prices on the chosen path, and otherwise we get nothing from the buyer. The objective of the problem is to find prices for the edges that maximizes the revenue obtained by selling the edges to the buyers who can afford their path. There is no restriction on the number of buyers that can simultaneously use an edge. In this paper, we consider the tollboth problem on a tree. Hence, each si, ti path is uniquely determined. A upward instance is when the tree is rooted at a vertex r, and ti is an ancestor of si for each buyer Pi, i = 1,..., m. We show that any solution to a naturally defined upward instance obtained from a tollbooth instance can be converted back to a solution to the orginial problem at a loss of at most a constant factor in the approximation. Using known results on the tollbooth problem for instances, we obtain improved algorithms for the tollbooth problem on trees.