Computing and Combinatorics

Optimal stopping theory considers the design of online algorithms for stopping a random sequence subject to an optimization criterion. For example, the famous secretary problem asks to identify a stopping rule that maximizes the probability of selecting the maximum element in a sequence presented in uniformly random order. In a similar vein, the prophet inequality of Krengel, Sucheston, and Garling establishes the existence of an online algorithm for selecting one element from a sequence of independent random numbers, such that the expected value of the chosen element is at least half the expectation of the maximum. A rich set of problems emerges when one combines these models with notions from combinatorial optimization by allowing the algorithm to select multiple elements from the sequence, subject to a combinatorial feasibility constraint on the set selected. A sequence of results during the past ten years have contributed greatly to our understanding of these problems. I will survey some of these developments and their applications to topics in algorithmic game theory. D.-Z. Du and G. Zhang (Eds.): COCOON 2013, LNCS 7936, p. 4, 2013. c © Springer-Verlag Berlin Heidelberg 2013 New Bounds for the Balloon Popping Problem Davide Bilò and Vittorio Bilò 1 Dipartimento di Teorie e Ricerche dei Sistemi Culturali, University of Sassari Piazza Conte di Moriana, 8, 07100 Sassari, Italy davide.bilo@uniss.it 2 Department of Mathematics and Physics “Ennio De Giorgi”, University of Salento Provinciale Lecce-Arnesano, P.O. Box 193, 73100 Lecce, Italy vittorio.bilo@unisalento.it 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) 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/6 + 1/4 ≈ 1.8949. 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) 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/6 + 1/4 ≈ 1.8949.