In the generalized caching problem, we have a set of pages and a cache of size $k$. Each page $p$ has a size $w_p\ge 1$ and fetching cost $c_p$ for loading the page into the cache. At any point in time, the sum of the sizes of the pages stored in the cache cannot exceed $k$. The input consists of a sequence of page requests. If a page is not present in the cache at the time it is requested, it has to be loaded into the cache incurring a cost of $c_p$. We give a randomized $O(\log k)$-competitive online algorithm for the generalized caching problem, improving the previous bound of $O(\log^2 k)$ by Bansal, Buchbinder, and Naor (STOC'08). This improved bound is asymptotically tight and of the same order as the known bounds for the classic problem with uniform weights and sizes. We follow the LP based techniques proposed by Bansal et al. and our main contribution are improved and slightly simplified methods for rounding fractional solutions online.
(From a paper at SODA 2012 by Anna Adamaszek, Artur Czumaj, Matthias Englert, and Harald Racke.)