Online Lower Bounds via Duality

TL;DR

This paper introduces a novel linear programming duality-based framework for deriving tight lower bounds on the competitive ratio of online algorithms across various problems, providing a unified and robust approach.

Contribution

It presents a general technique leveraging LP duality to systematically obtain online lower bounds, contrasting previous methods focused on positive results.

Findings

New tight lower bounds for Vector Bin Packing, Ad-auctions, and Capital Investment problems.

A unified LP-based approach simplifies and strengthens the derivation of online lower bounds.

The framework can reconstruct existing bounds and is applicable to many online problems.

Abstract

In this paper, we exploit linear programming duality in the online setting (i.e., where input arrives on the fly) from the unique perspective of designing lower bounds on the competitive ratio. In particular, we provide a general technique for obtaining online deterministic and randomized lower bounds (i.e., hardness results) on the competitive ratio for a wide variety of problems. We show the usefulness of our approach by providing new, tight lower bounds for three diverse online problems. The three problems we show tight lower bounds for are the Vector Bin Packing problem, Ad-auctions (and various online matching problems), and the Capital Investment problem. Our methods are sufficiently general that they can also be used to reconstruct existing lower bounds. Our techniques are in stark contrast to previous works, which exploit linear programming duality to obtain positive results, often via the useful primal-dual scheme. We design a general recipe with the opposite aim of obtaining negative results via duality. The general idea behind our approach is to construct a primal linear program based on a collection of input sequences, where the objective function corresponds to optimizing the competitive ratio. We then obtain the corresponding dual linear program and provide a feasible solution, where the objective function yields a lower bound on the competitive ratio. Online lower bounds are often achieved by adapting the input sequence according to an online algorithm's behavior and doing an appropriate ad hoc case analysis. Using our unifying techniques, we simultaneously combine these cases into one linear program and achieve online lower bounds via a more robust analysis. We are confident that our framework can be successfully applied to produce many more lower bounds for a wide array of online problems.