We obtain a near-tight bound of 0(n^{3 + \varepsilon }), for any \varepsilon > 0, on the complexity of the overlay of the minimization diagrams of two collections of surfaces in four dimensions. This settles a long-standing problem in the theory of arrangements, most recently cited by Agarwal and Sharir [3, Open Problem 2], and substantially improves and simplifies a result previously published by the authors [15]. Our bound has numerous algorithmic and combinatorial applications, some of which are presented in this paper.

Our result is obtained by introducing a new approach to the analysis of combinatorial structures arising in geometric arrangements of surfaces. This approach, which we call the ?partition technique?, is based on k-fold divide and conquer, in which a given collection F of n surfaces is partitioned into k subcollections F_i of {n \mathord{\left/ {\vphantom {n k}} \right. \kern-\nulldelimiterspace} k} surfaces each, and the complexity of the relevant combinatorial structure in F is recursively related to the complexities of the corresponding structures in each of the F_i?s. We introduce this approach by applying it first to obtain a new simple proof for the known near-quadratic bound on the complexity of an overlay of two minimization diagrams of collections of surfaces in \mathbb{R}^3, thereby simplifying the previously available proof [2].