Stochastic on-line knapsack problems
Mathematical Programming: Series A and B
Online computation and competitive analysis
Online computation and competitive analysis
Speed is as powerful as clairvoyance
Journal of the ACM (JACM)
Algorithmics for Hard Problems
Algorithmics for Hard Problems
Resource augmentation for online bounded space bin packing
Journal of Algorithms
Budget Constrained Bidding in Keyword Auctions and Online Knapsack Problems
WINE '08 Proceedings of the 4th International Workshop on Internet and Network Economics
On the Advice Complexity of Online Problems
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Online knapsack with resource augmentation
Information Processing Letters
Information complexity of online problems
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
Online computation with advice
Theoretical Computer Science
On the advice complexity of the k-server problem
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
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We study the advice complexity and the random bit complexity of the online knapsack problem: Given a knapsack of unit capacity, and n items that arrive in successive time steps, an online algorithm has to decide for every item whether it gets packed into the knapsack or not. The goal is to maximize the value of the items in the knapsack without exceeding its capacity. In the model of advice complexity of online problems, one asks how many bits of advice about the unknown parts of the input are both necessary and sufficient to achieve a specific competitive ratio. It is well-known that even the unweighted online knapsack problem does not admit any competitive deterministic online algorithm. We show that a single bit of advice helps a deterministic algorithm to become 2-competitive, but that Ω(log n) advice bits are necessary to further improve the deterministic competitive ratio. This is the first time that such a phase transition for the number of advice bits has been observed for any problem. We also show that, surprisingly, instead of an advice bit, a single random bit allows for a competitive ratio of 2, and any further amount of randomness does not improve this. Moreover, we prove that, in a resource augmentation model, i.e., when allowing a little overpacking of the knapsack, a constant number of advice bits suffices to achieve a near-optimal competitive ratio. We also study the weighted version of the problem proving that, with O(log n) bits of advice, we can get arbitrarily close to an optimal solution and, using asymptotically fewer bits, we are not competitive.