7,302 research outputs found

    Approximation Algorithms for the Max-Buying Problem with Limited Supply

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    We consider the Max-Buying Problem with Limited Supply, in which there are nn items, with CiC_i copies of each item ii, and mm bidders such that every bidder bb has valuation vibv_{ib} for item ii. The goal is to find a pricing pp and an allocation of items to bidders that maximizes the profit, where every item is allocated to at most CiC_i bidders, every bidder receives at most one item and if a bidder bb receives item ii then pivibp_i \leq v_{ib}. Briest and Krysta presented a 2-approximation for this problem and Aggarwal et al. presented a 4-approximation for the Price Ladder variant where the pricing must be non-increasing (that is, p1p2pnp_1 \geq p_2 \geq \cdots \geq p_n). We present an e/(e1)e/(e-1)-approximation for the Max-Buying Problem with Limited Supply and, for every ε>0\varepsilon > 0, a (2+ε)(2+\varepsilon)-approximation for the Price Ladder variant

    On Profit-Maximizing Pricing for the Highway and Tollbooth Problems

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    In the \emph{tollbooth problem}, we are given a tree \bT=(V,E) with nn edges, and a set of mm customers, each of whom is interested in purchasing a path on the tree. Each customer has a fixed budget, and the objective is to price the edges of \bT such that the total revenue made by selling the paths to the customers that can afford them is maximized. An important special case of this problem, known as the \emph{highway problem}, is when \bT is restricted to be a line. For the tollbooth problem, we present a randomized O(logn)O(\log n)-approximation, improving on the current best O(logm)O(\log m)-approximation. We also study a special case of the tollbooth problem, when all the paths that customers are interested in purchasing go towards a fixed root of \bT. In this case, we present an algorithm that returns a (1ϵ)(1-\epsilon)-approximation, for any ϵ>0\epsilon > 0, and runs in quasi-polynomial time. On the other hand, we rule out the existence of an FPTAS by showing that even for the line case, the problem is strongly NP-hard. Finally, we show that in the \emph{coupon model}, when we allow some items to be priced below zero to improve the overall profit, the problem becomes even APX-hard

    Sequential item pricing for unlimited supply

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    We investigate the extent to which price updates can increase the revenue of a seller with little prior information on demand. We study prior-free revenue maximization for a seller with unlimited supply of n item types facing m myopic buyers present for k < log n days. For the static (k = 1) case, Balcan et al. [2] show that one random item price (the same on each item) yields revenue within a \Theta(log m + log n) factor of optimum and this factor is tight. We define the hereditary maximizers property of buyer valuations (satisfied by any multi-unit or gross substitutes valuation) that is sufficient for a significant improvement of the approximation factor in the dynamic (k > 1) setting. Our main result is a non-increasing, randomized, schedule of k equal item prices with expected revenue within a O((log m + log n) / k) factor of optimum for private valuations with hereditary maximizers. This factor is almost tight: we show that any pricing scheme over k days has a revenue approximation factor of at least (log m + log n) / (3k). We obtain analogous matching lower and upper bounds of \Theta((log n) / k) if all valuations have the same maximum. We expect our upper bound technique to be of broader interest; for example, it can significantly improve the result of Akhlaghpour et al. [1]. We also initiate the study of revenue maximization given allocative externalities (i.e. influences) between buyers with combinatorial valuations. We provide a rather general model of positive influence of others' ownership of items on a buyer's valuation. For affine, submodular externalities and valuations with hereditary maximizers we present an influence-and-exploit (Hartline et al. [13]) marketing strategy based on our algorithm for private valuations. This strategy preserves our approximation factor, despite an affine increase (due to externalities) in the optimum revenue.Comment: 18 pages, 1 figur

    Optimal Real-Time Bidding Strategies

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    The ad-trading desks of media-buying agencies are increasingly relying on complex algorithms for purchasing advertising inventory. In particular, Real-Time Bidding (RTB) algorithms respond to many auctions -- usually Vickrey auctions -- throughout the day for buying ad-inventory with the aim of maximizing one or several key performance indicators (KPI). The optimization problems faced by companies building bidding strategies are new and interesting for the community of applied mathematicians. In this article, we introduce a stochastic optimal control model that addresses the question of the optimal bidding strategy in various realistic contexts: the maximization of the inventory bought with a given amount of cash in the framework of audience strategies, the maximization of the number of conversions/acquisitions with a given amount of cash, etc. In our model, the sequence of auctions is modeled by a Poisson process and the \textit{price to beat} for each auction is modeled by a random variable following almost any probability distribution. We show that the optimal bids are characterized by a Hamilton-Jacobi-Bellman equation, and that almost-closed form solutions can be found by using a fluid limit. Numerical examples are also carried out

    Exact algorithms for procurement problems under a total quantity discount structure.

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    In this paper, we study the procurement problem faced by a buyer who needs to purchase a variety of goods from suppliers applying a so-called total quantity discount policy. This policy implies that every supplier announces a number of volume intervals and that the volume interval in which the total amount ordered lies determines the discount. Moreover, the discounted prices apply to all goods bought from the supplier, not only to those goods exceeding the volume threshold. We refer to this cost-minimization problem as the TQD problem. We give a mathematical formulation for this problem and argue that not only it is NP-hard, but also that there exists no polynomial-time approximation algorithm with a constant ratio (unless P = NP). Apart from the basic form of the TQD problem, we describe three variants. In a first variant, the market share that one or more suppliers can obtain is constrained. Another variant allows the buyer to procure more goods than strictly needed, in order to reach a lower total cost. In a third variant, the number of winning suppliers is limited. We show that the TQD problem and its variants can be solved by solving a series of min-cost flow problems. Finally, we investigate the performance of three exact algorithms (min-cost flow based branch-and-bound, linear programming based branch-and-bound, and branch-and-cut) on randomly generated instances involving 50 suppliers and 100 goods. It turns out that even the large instances of the basic problem are solved to optimality within a limited amount of time. However, we find that different algorithms perform best in terms of computation time for different variants.Algorithms; Approximation; Branch-and-bound; Complexity; Cost; Exact algorithm; Intervals; Linear programming; Market; Min-cost flow; Order; Performance; Policy; Prices; Problems; Procurement; Reverse auction; Structure; Studies; Suppliers; Time; Volume discounts;

    Cookie Clicker

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    Cookie Clicker is a popular online incremental game where the goal of the game is to generate as many cookies as possible. In the game you start with an initial cookie generation rate, and you can use cookies as currency to purchase various items that increase your cookie generation rate. In this paper, we analyze strategies for playing Cookie Clicker optimally. While simple to state, the game gives rise to interesting analysis involving ideas from NP-hardness, approximation algorithms, and dynamic programming

    Lotsize optimization leading to a pp-median problem with cardinalities

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    We consider the problem of approximating the branch and size dependent demand of a fashion discounter with many branches by a distributing process being based on the branch delivery restricted to integral multiples of lots from a small set of available lot-types. We propose a formalized model which arises from a practical cooperation with an industry partner. Besides an integer linear programming formulation and a primal heuristic for this problem we also consider a more abstract version which we relate to several other classical optimization problems like the p-median problem, the facility location problem or the matching problem.Comment: 14 page
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