Advice Complexity of Priority Algorithms

TL;DR

This paper introduces a framework for establishing lower bounds on the advice complexity of priority algorithms, demonstrating its effectiveness through the Pair Matching problem and applications to various optimization problems.

Contribution

It develops a reduction-based framework for proving advice complexity lower bounds in the priority model, simplifying the process of analyzing algorithm limitations.

Findings

Established strong lower bounds for Pair Matching in the advice-augmented priority model.

Created a general reduction template applicable to multiple optimization problems.

Demonstrated the framework's utility by deriving bounds for standard discrete optimization problems.

Abstract

The priority model of "greedy-like" algorithms was introduced by Borodin, Nielsen, and Rackoff in 2002. We augment this model by allowing priority algorithms to have access to advice, i.e., side information precomputed by an all-powerful oracle. Obtaining lower bounds in the priority model without advice can be challenging and may involve intricate adversary arguments. Since the priority model with advice is even more powerful, obtaining lower bounds presents additional difficulties. We sidestep these difficulties by developing a general framework of reductions which makes lower bound proofs relatively straightforward and routine. We start by introducing the Pair Matching problem, for which we are able to prove strong lower bounds in the priority model with advice. We develop a template for constructing a reduction from Pair Matching to other problems in the priority model with advice -- this part is technically challenging since the reduction needs to define a valid priority function for Pair Matching while respecting the priority function for the other problem. Finally, we apply the template to obtain lower bounds for a number of standard discrete optimization problems.