On the Hardness of Pricing Loss-leaders

Journal of Revenue and Pricing Management, 2009

In this work, we establish a parallel between two classes of pricing problems that have attracted the attention of researchers in economics, theoretical computer science and operations research, each community addressing issues from its own vantage point. More precisely, we contract the problems of pricing a network or a product line, in order to achieve maximum revenue, given that customers maximize their individual utility. Throughout the paper, we focus on problems that can be formulated as mixed integer programs.

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, 2013

We resolve the complexity of revenue-optimal deterministic auctions in the unit-demand single-buyer Bayesian setting, i.e., the optimal item pricing problem, when the buyer's values for the items are independent. We show that the problem of computing a revenue-optimal pricing can be solved in polynomial time for distributions of support size 2, and its decision version is NP-complete for distributions of support size 3. We also show that the problem remains NP-complete for the case of identical distributions. * Columbia University.

Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, 2011

In the highway problem, we are given an n-edge line graph (the highway), and a set of paths (the drivers), each one with its own budget. For a given assignment of edge weights (the tolls), the highway owner collects from each driver the weight of the associated path, when it does not exceed the budget of the driver, and zero otherwise. The goal is choosing weights so as to maximize the profit. A lot of research has been devoted to this apparently simple problem. The highway problem was shown to be strongly NP-hard only recently [Elbassioni,Raman,Ray,Sitters-'09]. The bestknown approximation is O(log n/ log log n) [Gamzu,Segev-'10], which improves on the previous-best O(log n) approximation [Balcan,Blum-'06]. Better approximations are known for a number of special cases. Finding a constant (or better!) approximation algorithm for the general case is a challenging open problem.

Networks, 2009

We address the algorithmic complexity of a profit maximization problem in capacitated, undirected networks. We are asked to price a set of m capacitated network links to serve a set of n potential customers. Each customer is interested in purchasing a network connection that is specified by a simple path in the network and has a maximum budget that we assume to be known to the seller. The goal is to decide which customers to serve, and to determine prices for all network links in order to maximize the total profit. We address this pricing problem in different network topologies. More specifically, we derive several results on the algorithmic complexity of this profit maximization problem, given that the network is either a path, a cycle, a tree, or a grid. Our results include approximation algorithms as well as inapproximability results.

… Complexity (ECCC), TR09-112 (November 3, …

Consider a communication network represented by a directed graph G = (V, E) of n nodes and m edges. Assume that edges in E are partitioned into two sets: a set C of edges with a fixed non-negative real cost, and a set P of edges whose costs are instead priced by a leader. This is done with the final intent of maximizing a revenue that will be returned for their use by a follower, whose goal in turn is to select for his communication purposes a subnetwork of G minimizing a given objective function of the edge costs. In this paper, we study the natural setting in which the follower computes a single-source shortest paths tree of G, and then returns to the leader a payment equal to the sum of the selected priceable edges. Thus, the problem can be modeled as a one-round two-player Stackelberg Network Pricing Game, but with the novelty that the objective functions of the two players are asymmetric, in that the revenue returned to the leader for any of her selected edges is not equal to the cost of such an edge in the follower's solution. As is shown, for any > 0 and unless P = NP, the leader's problem of finding an optimal pricing is not approximable within n 1/2− , while, if G is unweighted and the leader can only decide which of her edges enter in the solution, then the problem is not approximable within n 1/3−. On the positive side, we devise a strongly polynomial-time O(n)-approximation algorithm, which favorably compares against the classic approach based on a single-price algorithm. Finally, motivated by practical applications, we consider the special cases in which edges in C are unweighted and happen to form two popular network topologies, namely stars and chains, and we provide a comprehensive characterization of their computational tractability.

2006

We consider a revenue maximization problem where we are selling a set of m items, each of which available in a certain quantity (possibly unlimited) to a set of n bidders. Bidders are single minded, that is, each bidder requests exactly one subset, or bundle of items. Each bidder has a valuation for the requested bundle that we assume to be known to the seller. The task is to find an envy-free pricing such as to maximize the revenue of the seller. We derive several complexity results and algorithms for several variants of this pricing problem. In fact, the settings that we consider address problems where the different items are 'homogeneous' in some sense. First, we introduce the notion of affine price functions that can be used to model situations much more general than the usual combinatorial pricing model that is mostly addressed in the literature. We derive fixed-parameter polynomial time algorithms as well as inapproximability results. Second, we consider the special case of combinatorial pricing, and introduce a monotonicity constraint that can also be seen as 'global' envy-freeness condition. We show that the problem remains strongly NP-hard, and we derive a PTAS -thus breaking the inapproximability barrier known for the general case. As a special case, we finally address the notorious highway pricing problem under the global envy-freeness condition.

4OR, 2011

We consider the problem of pricing items in order to maximize the revenue obtainable from a set of single minded customers. We relate the tractability of the problem to structural properties of customers' valuations: the problem admits an efficient approximation algorithm, parameterized along the inhomogeneity of the valuations.

Networks, 2012

In Stackelberg pricing a leader sets prices for items to maximize revenue from a follower purchasing a feasible subset of items. We consider computationally bounded followers who cannot optimize exactly over the range of all feasible subsets, but who apply publicly known algorithms to determine the items to purchase. This corresponds to general multidimensional pricing when customers cannot optimize their valuation functions efficiently but still aim to act rationally to the best of their ability. We consider two versions of this novel type of pricing problem. In the MIN-KNAPSACK variant items are weighted objects and the follower seeks to purchase a min-cost selection of objects of some bounded weight. When he uses a greedy 2-approximation algorithm, we provide a polynomial-time (2 + ε)-approximation algorithm for the leader's revenue maximization problem based on so-called near-uniform price assignments. We also prove the problem to be strongly NP-hard. In the SET-COVER variant items are subsets of some ground set which the follower seeks to cover. When he uses a standard primal-dual approach, we prove that exact revenue maximization is possible in polynomial time when

Journal of Combinatorial Optimization

In this paper, we consider extensions of the Maximum-Profit Public Transportation Route Planning Problem, or simply Maximum-Profit Routing Problem (MPRP), introduced in Armaselu and Daescu (Approximation algorithms for the maximum profit pick-up problem with time windows and capacity constraint, 2016. arXiv:1612.01038, Interactive assisting framework for maximum profit routing in public transportation in smart cities, PETRA, 13–16, 2017). Specifically, we consider MPRP with Time-Variable Supply (MPRP-VS), in which the quantity $$q_i(t)$$ q i ( t ) supplied at site i is linearly increasing in time t, as opposed to the original MPRP problem, where the quantity is constant in time. For MPRP-VS, our main result is a $$5.5 \log {T} (1 + \epsilon ) \left( 1 + \frac{1}{1 + \sqrt{m}}\right) ^2$$ 5.5 log T ( 1 + ϵ ) 1 + 1 1 + m 2 approximation algorithm, where T is the latest time window and m is the number of vehicles used. We also study the MPRP with Multiple Vehicles per Site, in which a ...

Siam Journal on Computing, 2011

We study the envy-free pricing problem faced by a profit maximizing seller when there is metric substitutability among the items-consumer i's value for item j is vi − ci,j, and the substitution costs, {ci,j}, form a metric. Our model is motivated from the observation that sellers often sell the same product at different prices in different locations, and rational consumers optimize the tradeoff between prices and substitution costs. While the general envy-free pricing problem is hard to approximate, the addition of metric substitutability constraints allows us to solve the problem exactly in polynomial time by reducing it to an instance of weighted independent set on a perfect graph. When the substitution costs do not form a metric, even in cases when a (1 +)-approximate triangle inequality holds, the problem becomes N P-hard. Our results show that triangle inequality is the exact sharp threshold for the problem of going from "tractable" to "hard". We then turn our attention to the multi-unit demand case, where consumers request multiple copies of the item. This problem has an interesting paradoxical non-monotonicity: The optimal revenue the seller can extract can actually decrease when consumers' demands increase. We show that in this case the revenue maximization problem becomes AP Xhard and give an O(log D) approximation algorithm, where D is the ratio of the largest to smallest demand. We extend these techniques to the more general case of arbitrary non-decreasing value functions, and give an O(log 3 D) approximation algorithm.