New bounds for the balloon popping problem

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 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 π 2/6 + 1/4 ≈ 1.8949. © 2013 Springer-Verlag Berlin Heidelberg.