When a store sells items to customers, the store wishes to determine the prices of the items to maximize its profit. Intuitively, if the store sells the items with low (resp. high) prices, the customers buy more (resp. less) items, which provides less profit to the store. So it would be hard for the store to decide the prices of items. Assume that the store has a set V of n items and there is a set E of m customers who wish to buy those items, and also assume that each item i \in V has the production cost d_i and each customer e_j \in E has the valuation v_j on the bundle e_j \subseteq V of items. When the store sells an item i \in V at the price r_i, the profit for the item i is p_i=r_i-d_i. The goal of the store is to decide the price of each item to maximize its total profit. In most of the previous works, the item pricing problem was considered under the assumption that p_i \geq 0 for each i \in V, however, Balcan, et al. [In Proc. of WINE, LNCS 4858, 2007] introduced the notion of loss-leader, and showed that the seller can get more total profit in the case that p_i < 0 is allowed than in the case that p_i < 0 is not allowed. In this paper, we consider the line and the cycle highway problem, and show approximation algorithms for the line and/or cycle highway problem for which the smallest valuation is s and the largest valuation is \ell or all valuations are identical.
When a store sells items to customers, the store wishes to decide the prices of items to maximize its profit. Intuitively, if the store sells the items with low (resp. high) prices, then the customers buy more (resp. less) items, which provides less profit to the store. So it would be hard for the store to decide the prices of items. Assume that the store has a set I = {1, 2, . . . , n} of n items and there is a set C = {c 1 , c 2 , . . . , c m } of m customers who wish to buy the items. The goal of the store is to decide the price of each item to maximize its profit. We refer to this problem as the item pricing problem. We classify the item pricing problem according to how many items the store can sell and how the customers valuate items. If the store can sell each item i with unlimited (resp. limited) amount, we refer to this as the unlimited (resp. limited) supply model. The item pricing problem is said to be single-minded [10] if each customer c j ∈ C is interested in only a single bundle e j = {j 1 , j 2 , . . . , } ⊆ I of items with valuation v j ≥ 0 and has valuation "0" on all other bundles of items. We say that the item pricing problem is unit-demand [10] if each customer c j ∈ C assigns valuation v i j ≥ 0 to each item i ∈ I and buys one of the most beneficial items for c j ∈ C. By regarding the set I of n items as the set V of n vertices and the set C of m customers as the set E of m hyperedges, each of which has weight v j ≥ 0, this can be formulated by a weighted hypergraph G = (V, E, {v j }). Note that the hypergraph G might have selfloops (corresponding to customers that are interested in a single item) and multiedges (corresponding to customers that want to get the same bundle of items). For a weighted hypergraph G = (V, E, {v j }), assume that each item i ∈ V has the production cost d i and each customer e j ∈ E has the valuation v j . For G, we define a reduced instance G = (V, E, {w j }) to be w j = v j -i∈e j d i for each e j ∈ E. If an item i ∈ V is assigned a price p i in the reduced instance G, then its selling price is given by r i = p i + d i . In this paper, we focus on the single-minded and unlimited supply model and consider reduced instances G's of weighted hypergraphs. We say that G = (V, E, {w j }) is an instance of the k-hypergraph vertex pricing problem if |e j | ≤ k for each e j ∈ E, an instance of the graph vertex pricing problem if |e j | ≤ 2 for each e j ∈ E, and an instance of the bipartite graph vertex pricing problem if G is a bipartite graph. As a special case of the hypergraph vertex pricing problem, we also say that G = (V, E, {w j }) is an instance of the highway problem if each e j ∈ E is an interval on V (the definition will be given in Definition 2.5 for the line highway problem and in Definition 2.6 for the cycle highway problem).
In most of the previous works [1,4,5,10], the item pricing problem is considered under the model that p i ≥ 0 for each item i ∈ V (this is called the positive price model). By introducing the notion of loss-leader [6], however, Balcan, et al. [3] consider several price models in which p i < 0 for some item i ∈ V (these are referred to as the discount model, the B-bounded discount model, the coupon model, etc., and are formally defined in Subsection 2.1), and showed that the seller could get more profit in the case that p i < 0 is allowed than in the case that p i < 0 is not allowed.
For the hypergraph vertex pricing problem, Guruswami, et al. [10,Theorem 5.2] show an O(log m+log n)approximation algorithm. 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 ǫ ) for some ǫ > 0. For the k-hypergraph vertex pricing problem, Briest and Krysta [4,Theorem 5.1] show an O(k 2 )-approximation algorithm, which is improved to an O(k)-approximation algorithm [1, Theorem 2]. For the graph vertex pricing problem, Balcan and Blum derive a 1/4-approximation algorithm [1, Theorem 1], while by the reduction from the vertex cover, Guruswami, et al. [10,Theorem 3.1] show that the graph vertex pricing problem is APX-hard even when all valuations are identical (if selfloops are allowed) or all valuations are either 1 or 2 (if selfloops are not allowed). For the highway problem, Balcan and Blum [1,Theorem 3] show an O(log n)-approximation algorithm and for the highway problem that forms a hierarchy, Balcan and Blum [1,Theorem 4] show a fully polynomial time approximation scheme. For the nonapproximability for the highway problem, see [4,9].
For the highway problem, we know the Ω(log n) gap between the positive price model and the (B-bounded) discount model [2, Theorem 1], and the Ω(log n) gap between the coupon model and the (B-bounded) discount model [2, Theorem 2]. For the graph vertex pricing problem, the Ω(log n) gap between the positive price model and the B-bounded discount model [2, Theorem
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