On-line analytical processing (OLAP) requires e cient processing of complex decision support queries over very large databases. It is well accepted that pre-computed data cubes can help reduce the response time of such queries dramatically. A very important design issue of an e cient OLAP system is therefore the choice of the right data cubes to materialize. We call this problem the data cube schema design problem. In this paper we show that the problem of nding an optimal data cube schema for an OLAP system with limited memory is NP-hard. As a more computationally e cient alternative, we propose a greedy approximation algorithm cMP and its variants. Algorithm cMP consists of two phases. In the rst phase, an initial schema consisting of all the cubes required to e ciently answer the user queries is formed. In the second phase, cubes in the initial schema are selectively merged to satisfy the memory constraint. We show that cMP is very e ective in prunning the search space for an optim...