Optimal projective algorithms for the list update problem

Abstract

The list update problem is a classical online problem, with an optimal competitive ratio that is still open, 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 TIMESTAMP. This and many other known algorithms, like MTF, are projective in the sense that they can be defined by only looking at any pair of list items at a time. Projectivity 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 their competitive ratio is never smaller than 1.6. Therefore, COMB is a best possible projective algorithm, and any better algorithm, if it exists, would need a non-projective approach.