The Advice Complexity of a Class of Hard Online Problems

Abstract

The advice complexity of an online problem is a measure of how much knowledge of the future an online algorithm needs in order to achieve a certain competitive ratio. Using advice complexity, we define the first online complexity class, AOC. The class includes independent set, vertex cover, dominating set, and several others as complete problems. AOC-complete problems are hard, since a single wrong answer by the online algorithm can have devastating consequences. For each of these problems, we show that log 1 + (c − 1) c−1 /c c n = Θ (n/c) bits of advice are necessary and sufficient (up to an additive term of O(log n)) to achieve a competitive ratio of c. on the upper bound by a factor of log c. For the remaining problems, no bounds on their advice complexity were previously known.

Weighted Online Problems with Advice

“…Definition 2 (AOC [7]). A problem, P, is in AOC (Asymmetric Online Covering) if it can be defined as follows: The input to an instance of P consists of a sequence of n requests, σ = r 1 , .…”

“…Recently in [7], the first complexity class for online problems, AOC, was introduced. The class consists of online problems that can be described in the following way: The input is a sequence of requests and each request must either be accepted or rejected.…”

“…This is an upper bound on the advice complexity of all problems in AOC. In [7], a list of problems including Independent Set, Vertex Cover, Dominating Set, and Set Cover were proven AOC-complete. The paper [1] studies a semi-online version of scheduling where it is allowed to keep several parallel schedules and choose the best schedule in the end.…”

“…-For AOC-complete maximization problems, the weighted version is not significantly harder than the unweighted version: For any maximization problem in AOC (this includes, e.g., Independent Set), the c-competitive algorithm given in [7] for the unweighted version of the problem can be converted into a (1 + ε)c-competitive algorithm for the weighted version using only O((log 2 n)/ε) additional advice bits. Thus, a (1 + ε)c-competitive algorithm using at most B(n, c) + O((log 2 n)/ε) bits of advice is obtained.…”

“…If the largest weight w max cannot be arbitrarily larger than the smallest weight w min , the c-competitive algorithm given in [7] for the unweighted version can be converted into a c(1 + ε)-competitive algorithm for the weighted versions using B(n, c) + O(log 2 n + log(log(w max /w min )/ε)) advice bits in total.…”