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New bounds for the balloon popping problem

Author

Listed:
  • Davide Bilò

    (University of Sassari)

  • Vittorio Bilò

    (University of Salento)

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 $$1.68$$ 1.68 and design an $$O(n^5)$$ O ( n 5 ) algorithm for computing upper bounds as a function of the number of bidders $$n$$ n . Our algorithm provides an experimental evidence that the correct upper bound is a constant smaller than $$2$$ 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 $$\pi ^2/6+1/4\approx 1.8949$$ π 2 / 6 + 1 / 4 ≈ 1.8949 .

Suggested Citation

  • Davide Bilò & Vittorio Bilò, 2015. "New bounds for the balloon popping problem," Journal of Combinatorial Optimization, Springer, vol. 29(1), pages 182-196, January.
    • Handle: RePEc:spr:jcomop:v:29:y:2015:i:1:d:10.1007_s10878-013-9696-7
    DOI: 10.1007/s10878-013-9696-7
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