The String Guessing Problem as a Method to Prove Lower Bounds on the Advice Complexity

Abstract

The advice complexity of an online problem describes the additional information both necessary and sufficient for online algorithms to compute solutions of a certain quality. In this model, an oracle inspects the input before it is processed by an online algorithm. Depending on the input string, the oracle prepares an advice bit string that is accessed sequentially by the algorithm. The number of advice bits that are read to achieve some specific solution quality can then serve as a fine-grained complexity measure. The main contribution of this paper is to study a powerful method for proving lower bounds on the number of advice bits necessary. To this end, we consider the string guessing problem as a generic online problem and show a lower bound on the number of advice bits needed to obtain a good solution. We use special reductions from string guessing to improve the best known lower bound for the online set cover problem and to give a lower bound on the advice complexity of the online maximum clique problem.

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].…”

“…The GMP problem [19] and the String Guessing Problem [10] both contain a core special case of guessing a binary sequence. We use their results to show that an online algorithm needs a linear number of bits of advice to achieve a competitive ratio better than 9/8 for bin packing.…”

“…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].…”

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.…”

Weighted Online Problems with Advice

“…The problem minASGk is based on the binary string guessing problem [2,11]. Binary string guessing is similar to asymmetric string guessing, except that any wrong guess (0 instead of 1 or 1 instead of 0) gives a cost of 1.…”