Abstract
We consider the market mechanism to sell two types of products, A and B, to a set of buyers \(I=\{1, 2, \ldots , n\}\). The amounts of products are \(m_A\) and \(m_B\) respectively. Each buyer i has his information including the budget, the preference and the utility function. On collecting the information from all buyers, the market maker determines the price of each product and allocates some amount of product to each buyer. The objective of the market maker is designing a mechanism to maximize the total utility of the buyers in satisfying the semi market equilibrium. In this paper, we show that this problem is NP-hard and give an iterative algorithm with the approximation ratio 1.5. Moreover, we introduce a PTAS for the problem, which is an (\(1+\epsilon \))-approximation algorithm with the running time \(O(2^{1/\epsilon }+n\log n)\) for any positive \(\epsilon \).
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Acknowledgements
This research is supported by Major Scientific and Technological Special Project of Guizhou Province (20183001), China’s NSFC Grants (Nos. 71361003, 71461003), Shenzhen Research Grant (Nos. KQJSCX20180330170311901, JCYJ 20180305180840138 and GGFW2017073114031767), Hong Kong GRF-17208019, Natural Science Foundations of Guizhou Province (No. [2018] 3002) and the Key Project of Hebei Education Department of China (ZD2019021).
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Li, P., Hua, Q., Hu, Z. et al. Approximation algorithms for the selling with preference. J Comb Optim 40, 366–378 (2020). https://doi.org/10.1007/s10878-020-00602-3
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DOI: https://doi.org/10.1007/s10878-020-00602-3