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 + n ^p) 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.