The partitioning of a rectangular grid graph with weighted vertices into p connected components such that the component of smallest weight is as heavy as possible (the max‐min problem) is considered. It is shown that the problem is NP‐hard for rectangles with at least three rows. A shifting algorithm is given which approximates the optimal solution. Bounds for the relative error are determined under a posteriori hypotheses. A further shifting algorithm is also given which allows for error estimates under a priori hypotheses and for asymptotic error estimates. A similar approach can be taken with the problem of finding the partition whose largest component is as small as possible (the min‐max problem). The case of rectangles with two rows has a polynomial algorithm and is dealt with in another paper. © 1998 John Wiley & Sons, Inc. Networks 32: 115–125, 1998