How to Sell a Graph: Guidelines for Graph Retailers

Abstract

We consider a profit maximization problem where we are asked to price a set of m items that are to be assigned to a set of n customers. The items can be represented as the edges of an undirected (multi)graph G, where an edge multiplicity larger than one corresponds to multiple copies of the same item. Each customer is interested in purchasing a bundle of edges of G, and we assume that each bundle forms a simple path in G. Each customer has a known budget for her respective bundle, and is interested only in that particular bundle. The goal is to determine item prices and a feasible assignment of items to customers in order to maximize the total profit. When the underlying graph G is a path, we derive a fully polynomial time approximation scheme, complementing a recent NP-hardness result. If the underlying graph is a tree, and edge multiplicities are one, we show that the problem is polynomially solvable, contrasting its APX-hardness for the case of unlimited availability of items. However, if the underlying graph is a grid, and edge multiplicities are one, we show that it is even NP-complete to approximate the maximum profit to within a factor n 1−ε .

Approximation Algorithms for Single-minded Envy-free Profit-maximization Problems with Limited Supply

“…Part (i) of Theorem 1.1 answers positively one of the main open questions in Guruswami et al [13], of whether there exists a sublinear factor polynomial time approximation algorithm for the single-parameter limited-supply case. For general SMEFP, our approximation guarantee is close to the best possible since a simple reduction from the setpacking or independent-set problem shows that approximating general SMEFP within a factor better than m 1 2 or n is NP-hard even when u max = 1 [14]. Previously, for general SMEFP, and even for special cases such as the highway problem, the only known guarantees were either poly-time logarithmic guarantees (in m and/or n) for the unlimited-supply setting [13,3,5,4], quasipolynomial and pseudopolynomial algorithms, or approximation schemes for restricted instances [13,15,5,14].…”

“…Balcan and Blum [3] and independently Briest and Krysta [5] gave an FPTAS for the problem when the customer-paths form a laminar family (such an instance also captures an instance of the general problem with a laminar customer-subset family). For the limited-supply highway problem, Grigoriev et al [14] obtained an FPTAS under the assumption that the maximum supply is bounded, and recently Elbassioni et al [11] devised a quasi-PTAS. It is not clear if the rounding ideas in such schemes can be applied to the envy-free problem since envy-freeness imposes both upper-bound (for winners) and lower-bound (for losers) constraints on the price of a customer-subset, so rounding may not preserve the feasibility and profit of the solution.…”

Approximation Preserving Reductions among Item Pricing Problems

“…On the other hand, Demaine, et al [7,Theorem 3.2] present that it is hard to approximate the hypergraph vertex pricing problem within a factor of log δ n for some δ > 0 under the assumption that NP BPTIME(2 n ) for some > 0. [4], [8]. …”

Online pricing for bundles of multiple items

“…Some researchers have studied more complex and realistic user preferences in the complete information setting, with the aim of devising fast algorithms to give the optimal, or a good approximate, pricing solution [5][6][7][8][9][12][13][14][15][16][17][18][19]. Other researchers have studied the situation where the seller must set prices without knowledge, or complete knowledge, of the valuations of future users, with the aim of devising algorithms that will perform well even in the face of adversarial users trying to undermine the seller [1][2][3][4]10,11,20].…”

“…Both multiple types of items and single type of items have been considered. Previous work has mainly focused on two supply models: the unlimited supply model [2,5,9,15,17] where the number of each type of item is unbounded and the limited supply model [2,[6][7][8]12,16,18,19] where the number of each type of item is bounded by some value. As for the users, there are several users' behaviors studied, including single-minded [12][13][14][15]17,19] (each user is interested only in a particular set of items), unit-demand [5][6][7][8][9]17,19] (each user will buy at most one item in total) and envy free [2,8,12,15,17] (after the assignment, no user would prefer to be assigned a different set of items with the designated prices, loosely speaking, each user is happy with his/her purchase).…”