Approximation algorithms for array partitioning problems

We study the problem of optimally partitioning a two-dimensional array of elements by cutting each coordinate axis into p (respectively, q) intervals, resulting in p×q rectangular regions. This problem arises in several applications in databases, parallel computation, and image processing. Our main contribution are new approximation algorithms for these NP-complete problems that improve significantly over previously known bounds. The algorithms are fast and simple, work for a variety of measures of partitioning quality, generalize to dimensions d>2, and achieve almost optimal approximation ratios. We also extend previous NP-completeness results for this class of problems.