Towards the randomized k-server conjecture: A primal-dual approach

Recently, Coté et al. [10] proposed an approach for solving the k-server problem on Hierchically Separated Trees (HSTs). In particular, they define a problem on a uniform metric, and show that if an algorithm with a certain refined guarantee exists for it, then one can obtain polylogarithmic (in diameter) competitive factors for the k-server problem on HSTs by solving this problem recursively. By designing such an algorithm for a two point metric, they obtained a logarithmic competitive algorithm for well-separated binary HSTs. Extending their result to uniform metrics on arbitrarily many points would imply a poly-logarithmic competitive algorithm for k-server on general HSTs (and hence general metrics) and is thus of major interest. Here, we design such an algorithm for any uniform metric, provided the instance satisfies a certain "convexity" property. Even though this does not give a result for k-server, convexity seems to be a very natural property, and we give evidence that instances arising in the Coté et al. [10] reduction from k-server essentially possess this property, suggesting that this might be a promising approach. Already, our setting is general enough to model the finely competitive paging problem proposed by Blum et al. [4], who motivated it as a first step towards achieving a polylog(k) competitive algorithm for k-server. Our result implies an r + O(log k)-competitive algorithm for finely competitive paging, resolving the main open problem of [4]. Our results are based on an extension of the primaldual framework for online algorithms developed by Buchbinder and Naor [7]. The original approach works for problems whose offline version can be expressed as a packing or a covering linear program, possibly with box constraints. The online nature of the problem is modeled by revealing the constraints one by one and the requirement that variables can only be increased over time. Here, we consider more general types of constraints, where terms can be both positive and negative. Moreover, we allow the variables to both increase and decrease. This versatility allows us to model problems such as predicting with expert advice, which could not be modeled earlier. To show the simplicity and generality of this approach, we give an alternate O(log k)-competitive algorithm for weighted paging with a very simple proof. We also give an alternate primal-dual approach to design regret minimization algorithms for the problem of online prediction with expert advice. Our results suggest the possibility of a more general primal-dual framework for online problems beyond covering and packing LPs.