On the Approximability of Geometric and Geographic Generalization and the Min-Max Bin Covering Problem

We study the problem of abstracting a table of data about individuals so that no selection query can identify fewer than k individuals. As is common in existing work on this k-anonymization problem, the means we investigate to perform this anonymization is to generalize values of quasi-identifying attributes into equivalence classes. Since such data tables are intended for use in data mining, we consider the natural optimization criterion of minimizing the maximum size of any equivalence class, subject to the constraint that each is of size at least k. We show that it is impossible to achieve arbitrarily good polynomial-time approximations for a number of natural variations of the generalization technique, unless P = NP , even when the table has only a single quasi-identifying attribute that represents a geographic or unordered attribute: • Zip-codes: nodes of a planar graph generalized into connected subgraphs • GPS coordinates: points in R generalized into non-overlapping rectangles • Unordered data: text labels that can be grouped arbitrarily. These hard single-attribute instances of generalization problems contrast with the previously known NP-hard instances, which require the number of attributes to be proportional to the number of individual records (the rows of the table). In addition to impossibility results, we provide approximation algorithms for these difficult single-attribute generalization problems, which, of course, apply to multiple-attribute instances with one that is quasi-identifying. We show theoretically and experimentally that our approximation algorithms can come reasonably close to optimal solutions. Incidentally, the generalization problem for unordered data can be viewed as a novel type of bin packing problem—min-max bin covering—which may be of independent interest.