A stab at approximating minimum subadditive join

  • Authors:
  • Staal A. Vinterbo

  • Affiliations:
  • Decision Systems Group, Brigham and Women's Hospital, Boston and Harvard Medical School, Boston and Harvard-MIT, Division of Health Sciences and Technology, Boston

  • Venue:
  • WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
  • Year:
  • 2007

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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.