Abstract
The winner determination problem (WDP) arises in combinatorial auctions. It is known to be NP-hard. In this paper, we propose a discrete dynamic convexized method for solving this problem. We first propose an adaptive penalty function to convert the WDP into an equivalent unconstrained integer programming problem. Based on the structure of the WDP, we construct an unconstrained auxiliary function, which is maximized iteratively using a local search and is updated whenever a better maximizer is found. By increasing the value of a parameter in the auxiliary function, the maximization of the auxiliary function can escape from previously converged local maximizers. To evaluate the performance of the dynamic convexized method, extensive experiments were carried out on realistic test sets from the literature. Computational results and comparisons show that the proposed algorithm improved the best known solutions on a number of benchmark instances.
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Notes
\(ZPP\) is the class of problems that can be solved in polynomial time with randomized algorithm with zero probability of error.
For the presentation of the local search in this section we use \(\psi (x,R)\) in (4), instead of the auxiliary function. Description of the local search using the auxiliary function is a trivial adjustment of this.
According to Remark 3, suffice it to find a local maximizer of \((UP)\).
Note that the computational results might be improved by special tuning of the parameters on different instances. However, we used the parameters setting in Table 2 on all tested instances.
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Acknowledgments
The authors thank Dalila Boughaci for providing the data of the test instances. This research was supported partially by the National Natural Science Foundation of China under Grants 11301255, and 61170308, and the Science and Technology Project of the Education Bureau of Fujian, China, under Grant JA13246. The authors also thank anonymous reviewers whose comments and suggestions helped for improving the quality of the manuscript greatly.
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Lin, G., Zhu, W. & Ali, M.M. An effective discrete dynamic convexized method for solving the winner determination problem. J Comb Optim 32, 563–593 (2016). https://doi.org/10.1007/s10878-015-9883-9
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DOI: https://doi.org/10.1007/s10878-015-9883-9