In the offline list update problem, we maintain an unsorted linear list used as a dictionary. Accessing the item at position i in the list costs i units. In order to reduce access cost, we are allowed to update the list at any time by transposing consecutive items at a cost of one unit. Given a sequence of requests one has to serve in turn, we are interested in the minimal cost needed to serve all requests. Little is known about this problem. The best algorithm so far needs exponential time in the number of items in the list. We show that there is no polynomial algorithm unless P = NP.

The optimal competitive ratio for a randomized online list update algorithm is known to be at least 1.5 and at most 1.6, but the remaining gap is not yet closed. We present a new lower bound of 1.50084 for the partial cost model. The construction is based on game trees with incomplete information, which seem to be generally useful for the competitive analysis of online algorithms.

The optimal competitive factor for a randomized online list update algorithm is known to be at least 1.5 and at most 1.6, but the remaining gap is not yet closed. We present a new lower bound of 1.5003 under the simplifying assumption that the online algorithm may not use paid exchanges and for the partial cost model. The construction is based on game trees with incomplete information, which seem to be generally useful for the competitive analysis of online algorithms.

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.