A stab at approximating minimum subadditive join

Abstract

Let (L, *) be a semilattice, and let c: L → [0, ∞) be monotone and increasing on L. We state the Minimum Join problem as: given size n sub-collection X of L and integer k with 1 ≤ k ≤ n, find a size k sub-collection (x′1, x′2,..., x′k) of X that minimizes c(x′1 * x′2 * ⋯ * x′k). If c(a * b) ≤ c(a) + c(b) holds, we call this the Minimum Subadditive Join (MSJ) problem and present a greedy (k - p + 1)-approximation algorithm requiring O((k - p)n + np) joins for constant integer 0 < p ≤ k. We show that the MSJ Minimum Coverage problem of selecting k out of n finite sets such that their union is minimal is essentially as hard to approximate as the Maximum Balanced Complete Bipartite Subgraph (MBCBS) problem. The motivating by-product of the above is that the privacy in databases related k-ambiguity problem over L with subadditive information loss can be approximated within k - p, and that the k-ambiguity problem is essentially at least as hard to approximate as MBCBS. © Springer-Verlag Berlin Heidelberg 2007.