Exact and approximate discrete optimization algorithms for finding useful disjunctions of categorical predicates in data analysis

We discuss a discrete optimization problem that arises in data analysis from the binarization of categorical attributes. It can be described as the maximization of a function F (l1(x), l2(x)), where l1(x) and l2(x) are linear functions of binary variables x ∈ {0, 1}n, and F : R2 −→ R. Though this problem is NP-hard, in general, an optimal solution x∗ of it can be found, under some mild monotonicity conditions on F , in pseudo-polynomial time. We also present an approximation algorithm which finds an approximate binary solution x2, for any given 2 > 0, such that F (l1(x∗), l2(x∗)) − F (l1(x), l2(x)) < 2, at the cost of no more than O(n log n + 2C/ √ 2n) operations. Though in general C depends on the problem instance, for the problems arising from binarization of categorical variables it depends only on F , and for all functions considered we have C ≤ 1/√2. Acknowledgements: This research was partially supported by the National Science Foundation, Grant IIS-0118635, by the Office of Naval Research, Grant N00014-92-J-1375, and by the Rutgers Distributed Laboratory for Digital Libraries, a Strategic Opportunity Project of Rutgers, the State University of New Jersey.