Weighted Online Problems with Advice

TL;DR

This paper explores the advice complexity of weighted online problems within the AOC class, revealing differences between minimization and maximization, and extending techniques to broader problems like matching and scheduling.

Contribution

It introduces weighted versions of AOC problems, analyzes their advice complexity, and extends algorithmic techniques to non-complete and non-AOC problems.

Findings

Advice complexity differs significantly between weighted minimization and maximization problems.

Weighted algorithms outperform general AOC results for maximum matching.

Techniques are applicable to scheduling and other non-AOC problems.

Abstract

Recently, the first online complexity class, AOC, was introduced. The class consists of many online problems where each request must be either accepted or rejected, and the aim is to either minimize or maximize the number of accepted requests, while maintaining a feasible solution. All AOC-complete problems (including Independent Set, Vertex Cover, Dominating Set, and Set Cover) have essentially the same advice complexity. In this paper, we study weighted versions of problems in AOC, i.e., each request comes with a weight and the aim is to either minimize or maximize the total weight of the accepted requests. In contrast to the unweighted versions, we show that there is a significant difference in the advice complexity of complete minimization and maximization problems. We also show that our algorithmic techniques for dealing with weighted requests can be extended to work for non-complete AOC problems such as maximum matching (giving better results than what follow from the general AOC results) and even non-AOC problems such as scheduling.