The online knapsack problem: Advice and randomization

Abstract

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.For this problem, we show that a single bit of advice helps a deterministic online 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. Additionally, we 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 the online algorithm to overpack the knapsack by some small amount, 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. Furthermore, we show that an arbitrary number of random bits does not permit a constant competitive ratio.

Online Bin Packing with Advice

“…We refer to this model as advice-on-tape model. Since its introduction, the advice-on-tape model has been used to analyze the advice complexity of many online problems including paging [12,26,29], disjoint path allocation [12], job shop scheduling [12,29], k-server [11,30], knapsack [9], various coloring problems [5,21,7,32], set cover [28,10], maximum clique [10], and graph exploration [17].…”

“…Provided with the appropriate advice, the online algorithms are expected to achieve improved competitive ratios. The advice model has received significant attention since its introduction [12,26,19,11,29,30,9,6,17,21,28,10,7,32].…”

“…There is generally a trade-off between the number of advice bits and the quality of solutions. For some problems, including list updating and a variant of knapsack, a constant number of bits of advice is known to be sufficient to outperform all online algorithms [13,9]. On the other hand, for many online problems, at least a linear number of bits is required to achieve optimal solutions.…”

On the advice complexity of the k-server problem

“…Emek et al [13] proposed a different way to revise the model from Dobrev et al by restricting the online algorithm to read a fixed number of advice bits in every time step. With such an approach, however, it is not feasible to analyze a sublinear advice complexity (or even linear advice βn for β < 1), which is a serious issue for many online problems [3,4,5,7,14,20]. It is easy to simulate the model from Emek et al with our model, which is more general in this sense, and all lower bounds in our model directly carry over to their model.…”

Online multi-coloring with advice

“…Many other problems have been studied in an advice setting, including disjoint path allocation by Barhum et al [2], and job shop scheduling by Böckenhauer et al [6], as well as k-server by Böckenhauer et al [5], knapsack by Böckenhauer et al [7], set cover by Komm, Královič, and Mömke [29], metrical task systems by Emek et al [22], and buffer management by Dorrigiv, He, and Zeh [20].…”