Computational aspects of combinatorial pricing problems

Combinatorial pricing encompasses a wide range of natural optimization problems that arise in the computation of revenue maximizing pricing schemes for a given set of goods, as well as the design of profit maximizing auctions in strategic settings. We consider the computational side of several different multi-product and network pricing problems and, as most of the problems in this area are NP-hard, we focus on the design of approximation algorithms and corresponding inapproximability results. In the unit-demand multi-product pricing problem it is assumed that each consumer has different budgets for the products she is interested in and purchases a single product out of her set of alternatives. Depending on how consumers choose their products once prices are fixed we distinguish the min-buying, max-buying and rank-buying models, in which consumers select the affordable product with smallest price, highest price or highest rank according to some predefined preference list, respectively. We prove that the max-buying model allows for constant approximation guarantees and this is true even in the case of limited product supply. For the min-buying model we prove inapproximability beyond the known logarithmic guarantees under standard complexity theoretic assumptions. Surprisingly, this result even extends to the case of pricing with a price ladder constraint, i.e., a predefined relative order on the product prices. Furthermore, similar results can be shown for the uniform-budget version of the problem, which corresponds to a special case of the unit-demand envy-free pricing problem, under an assumption about the average case hardness of refuting random 3SAT-instances. Introducing the notion of stochastic selection rules we show that among a large class of selection rules based on the order of product prices the maxbuying model is in fact the only one allowing for sub-logarithmic approximation guarantees. In the single-minded pricing problem each consumer is interested in a single set of products, which she purchases if the sum of prices does not exceed her budget. It turns out that our results on envyfree unit-demand pricing can be extended to this scenario and yield inapproximability results for ratios expressed in terms of the number of distinct products, thereby complementing existing hardness results. On the algorithmic side, we present an algorithm with approximation guarantee that depends only on the maximum size of the sets and the number of requests per product. Our algorithm’s ratio matches previously known results in the worst case but has significantly better provable performance guarantees on sparse problem instances. Viewing single-minded as a network pricing problem in which we assign prices to edges and consumers want to purchase paths in the network, it is proven that the problem remains APX-hard even on extremely sparse instances. For the special case of pricing on a line with paths that are nested, we design an FPTAS and prove NP-hardness. In a Stackelberg network pricing game a so-called leader sets the prices on a subset of the edges of a network, the remaining edges have associated fixed costs. Once prices are fixed, one or more followers purchase min-cost subnetworks according to their requirements and pay the leader for all pricable edges contained in their networks. We extend the analysis of the known single-price algorithm, which assigns the same price to all pricable edges, from cases in which the feasible subnetworks of a follower form the basis of a matroid to the general case, thus, obtaining logarithmic approximation guarantees for general Stackelberg games. We then consider a special 2-player game in which the follower buys a min-cost vertex cover in a bipartite graph and the leader sets prices on a subset of the vertices. We prove that this problem is polynomial time solvable in some cases and allows for constant approximation guarantees in general. Finally, we point out that results on unit-demand and single-minded pricing yield several strong inapproximability results for Stackelberg pricing games with multiple followers

Packing Cars into Narrow Roads: PTASs for Limited Supply Highway

In the Highway problem, we are given a path with n edges (the highway), and a set of m drivers, each one characterized by a subpath and a budget. For a given assignment of edge prices (the tolls), the highway owner collects from each driver the total price of the associated path when it does not exceed drivers\u27s budget, and zero otherwise. The goal is to choose the prices to maximize the total profit. A PTAS is known for this (strongly NP-hard) problem [Grandoni,Rothvoss-SODA\u2711, SICOMP\u2716]. In this paper we study the limited supply generalization of Highway, that incorporates capacity constraints. Here the input also includes a capacity u_e >= 0 for each edge e; we need to select, among drivers that can afford the required price, a subset such that the number of drivers that use each edge e is at most u_e (and we get profit only from selected drivers). To the best of our knowledge, the only approximation algorithm known for this problem is a folklore O(log m) approximation based on a reduction to the related Unsplittable Flow on a Path problem (UFP). The main result of this paper is a PTAS for limited supply highway. As a second contribution, we study a natural generalization of the problem where each driver i demands a different amount d_i of capacity. Using known techniques, it is not hard to derive a QPTAS for this problem. Here we present a PTAS for the case that drivers have uniform budgets. Finding a PTAS for non-uniform-demand limited supply highway is left as a challenging open problem

Optimal Bundle Pricing for Homogeneous Items

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 affne 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.operations research and management science;

Price Strategy Implementation

Consider a situation in which a company sells several different items to a set of customers. However, the company is not satisfied with the current pricing strategy and wishes to implement new prices for the items. Implementing these new prices in one single step mightnot be desirable, for example, because of the change in contract prices for the customers. Therefore, the company changes the prices gradually, such that the prices charged to a subset of the customers, the target market, do not differ too much from one period to the next. We propose a polynomial time algorithm to implement the new prices in the minimum number of time periods needed, given that the prices charged to the customers in the target market increase by at most a factor 1 + δ, for predetermined δ > 0. Furthermore, we address the problem to maximize the revenue when also a maximum number of time periods is predetermined. For this problem, we describe a dynamic program if the numberof possible prices is limited, and a local search algorithm if all prices are allowed. Also, we present the integer program that models this problem. Finally, we apply the obtained algorithms in a practical study.operations research and management science;

Stackelberg Network Pricing Games

We study a multi-player one-round game termed Stackelberg Network Pricing Game, in which a leader can set prices for a subset of $m$ priceable edges in a graph. The other edges have a fixed cost. Based on the leader's decision one or more followers optimize a polynomial-time solvable combinatorial minimization problem and choose a minimum cost solution satisfying their requirements based on the fixed costs and the leader's prices. The leader receives as revenue the total amount of prices paid by the followers for priceable edges in their solutions, and the problem is to find revenue maximizing prices. Our model extends several known pricing problems, including single-minded and unit-demand pricing, as well as Stackelberg pricing for certain follower problems like shortest path or minimum spanning tree. Our first main result is a tight analysis of a single-price algorithm for the single follower game, which provides a $(1+\epsilon) \log m$-approximation for any $\epsilon >0$. This can be extended to provide a $(1+\epsilon)(\log k + \log m)$-approximation for the general problem and $k$ followers. The latter result is essentially best possible, as the problem is shown to be hard to approximate within \mathcal{O(\log^\epsilon k + \log^\epsilon m). If followers have demands, the single-price algorithm provides a $(1+\epsilon)m^2$-approximation, and the problem is hard to approximate within \mathcal{O(m^\epsilon) for some $\epsilon >0$. Our second main result is a polynomial time algorithm for revenue maximization in the special case of Stackelberg bipartite vertex cover, which is based on non-trivial max-flow and LP-duality techniques. Our results can be extended to provide constant-factor approximations for any constant number of followers

On Profit-Maximizing Pricing for the Highway and Tollbooth Problems

In the \emph{tollbooth problem}, we are given a tree \bT=(V,E) with $n$ edges, and a set of $m$ 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 $O(\log n)$-approximation, improving on the current best $O(\log m)$-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 $(1-\epsilon)$-approximation, for any $\epsilon > 0$, 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

On the Complexity of the Highway Pricing Problem

The highway pricing problem asks for prices to be determined for segments of a single highway such as to maximize the revenue obtainable from a given set of customers with known valuations. The problem is (weakly) NP-hard and a recent quasi-PTAS suggests that a PTAS might be in reach. Yet, so far it has resisted any attempt for constant-factor approximation algorithms. We relate the tractability of the problem to structural properties of customers' valuations. We show that the problem becomes NP-hard as soon as the average valuations of customers are not homogeneous, even under further restrictions such as monotonicity. Moreover, we derive an efficient approximation algorithm, parameterized along the inhomogeneity of customers' valuations. Finally, we discuss extensions of our results that go beyond the highway pricing problem.\u

Optimal Bundle Pricing with Monotonicity Constraint

We consider the problem to price (digital) items in order to maximize the revenue obtainable from a set of bidders. We suggest a natural monotonicity constraint on bundle prices, show that the problem remains NP-hard, and we derive a PTAS. We also discuss a special case, the highway pricing problem.operations research and management science;