Abstract
We reconsider the balloon popping problem, an intriguing combinatorial problem introduced in order to bound the competitiveness of ascending auctions with anonymous bidders with respect to the best fixed-price scheme. Previous works show that the optimal solution for this problem is in the range (1.6595,2). We give a new lower bound of \(1.68\) and design an \(O(n^5)\) algorithm for computing upper bounds as a function of the number of bidders \(n\). Our algorithm provides an experimental evidence that the correct upper bound is a constant smaller than \(2\), thus disproving a currently believed conjecture, and can be used to test the validity of a new conjecture we propose, according to which the upper bound would decrease to \(\pi ^2/6+1/4\approx 1.8949\).
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Acknowledgments
This work was partially supported by the PRIN 2010–2011 research project ARS TechnoMedia: “Algorithmics for Social Technological Networks” funded by the Italian Ministry of University.
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Bilò, D., Bilò, V. New bounds for the balloon popping problem. J Comb Optim 29, 182–196 (2015). https://doi.org/10.1007/s10878-013-9696-7
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DOI: https://doi.org/10.1007/s10878-013-9696-7