How Much Information about the Future Is Needed?

Abstract

We propose a new way of characterizing the complexity of online problems. Instead of measuring the degradation of output quality caused by the ignorance of the future we choose to quantify the amount of additional global information needed for an online algorithm to solve the problem optimally. In our model, the algorithm cooperates with an oracle that can see the whole input. We define the advice complexity of the problem to be the minimal number of bits (normalized per input request, and minimized over all algorithm-oracle pairs) communicated between the algorithm and the oracle in order to solve the problem optimally. Hence, the advice complexity measures the amount of problem-relevant information contained in the input.We introduce two modes of communication between the algorithm and the oracle based on whether the oracle offers an advice spontaneously (helper) or on request (answerer). We analyze the Paging and DiffServ problems in terms of advice complexity and deliver tight bounds in both communication modes.

Online Makespan Scheduling with Sublinear Advice

“…The advice complexity of online problems was first introduced by Dobrev, Královič and Pardubská in 2008 [4]. In their work, they used a different model.…”

“…In such a setting, merely one bit of advice suffices to achieve a competitive ratio of 4 3 : The oracle indicates whether there is a 2 3 -large job or not. The algorithm then works like the one described in the proof of Theorem 3.1, but recognizes the 2 3 -large job on its own without reading any index from the advice tape.…”

“…As a remark, we want to point out that scheduling this k-th job on one machine and all other jobs on the other machine would lead to a competitive ratio that is smaller than 4 3 (cf. Theorem 3.1 and its proof).…”

“…First, we give linear upper and lower bounds on the advice complexity for optimal algorithms with advice, with the bounds being tight up to one bit. Afterwards, we introduce a very simple online algorithm with logarithmic advice complexity and a competitive ratio of 4 3 . We also show that with a more involved strategy an online algorithm can get arbitrarily close to this competitive ratio using only a constant number of advice bits.…”

Advice Complexity of Fine-Grained Job Shop Scheduling

“…In 2008, Dobrev, Královič, and Pardubská introduced the concept of advice complexity for online problems [5]. Basically, they asked the question we asked at the beginning of this chapter.…”