Graph Pricing with Limited Supply
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We study approximation algorithms for graph pricing with vertex capacities yet without the traditional envy-free constraint. Specifically, we have a set of items V and a set of customers X where each customer i ∈ X has a budget b_i and is interested in a bundle of items S_i ⊆ V with |S_i| ≤ 2. However, there is a limited supply of each item: we only have μ_v copies of item v to sell for each v ∈ V. We should assign prices p(v) to each v ∈ V and chose a subset Y ⊆ X of customers so that each i ∈ Y can afford their bundle (p(S_i) ≤ b_i) and at most μ_v chosen customers have item v in their bundle for each item v ∈ V. Each customer i ∈ Y pays p(S_i) for the bundle they purchased: our goal is to do this in a way that maximizes revenue. Such pricing problems have been studied from the perspective of envy-freeness where we also must ensure that p(S_i) ≥ b_i for each i ∉ Y. However, the version where we simply allocate items to customers after setting prices and do not worry about the envy-free condition has received less attention. Our main result is an 8-approximation for the capacitated case via local search and a 7.8096-approximation in simple graphs with uniform vertex capacities. The latter is obtained by combing a more involved analysis of a multi-swap local search algorithm for constant capacities and an LP-rounding algorithm for larger capacities. If all capacities are bounded by a constant C, we further show a multi-swap local search algorithm yields an (4 ·2C-1/C + ϵ)-approximation. We also give a (4+ϵ)-approximation in simple graphs through LP rounding when all capacities are very large as a function of ϵ.
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