Optimal Lower Bounds for Projective List Update Algorithms

TL;DR

This paper proves that within the partial cost model, no projective list update algorithm can have a competitive ratio better than 1.6, establishing COMB as optimal among such algorithms.

Contribution

The paper characterizes all projective list update algorithms and shows their competitive ratio cannot be below 1.6, confirming the optimality of COMB in this class.

Findings

All projective algorithms have a competitive ratio ≥ 1.6.

COMB is proven to be optimal among projective algorithms in the partial cost model.

The lower bound matches the competitive ratio of the best known algorithm, COMB.

Abstract

The list update problem is a classical online problem, with an optimal competitive ratio that is still open, known to be somewhere between 1.5 and 1.6. An algorithm with competitive ratio 1.6, the smallest known to date, is COMB, a randomized combination of BIT and the TIMESTAMP algorithm TS. This and almost all other list update algorithms, like MTF, are projective in the sense that they can be defined by looking only at any pair of list items at a time. Projectivity (also known as "list factoring") simplifies both the description of the algorithm and its analysis, and so far seems to be the only way to define a good online algorithm for lists of arbitrary length. In this paper we characterize all projective list update algorithms and show that their competitive ratio is never smaller than 1.6 in the partial cost model. Therefore, COMB is a best possible projective algorithm in this model.