On the Approximability of the Minimum Test Collection Problem

Extended Abstract
  • Bjarni V. Halldórsson
  • Magnús M. Halldórsson
  • R. Ravi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2161)

Abstract

The minimum test collection problem is defined as follows. Given a ground set \( \mathcal{S} \) and a collection \( \mathcal{C} \) of tests (subsets of \( \mathcal{S} \)), find the minimum subcollection \( \mathcal{C}' \) of \( \mathcal{C} \) such that for every pair of elements (x, y) in \( \mathcal{S} \) there exists a test in \( \mathcal{C}' \) that contains exactly one of x and y. It is well known that the greedy algorithm gives a 1 + 2lnn approximation for the test collection problem where \( n = \left| \mathcal{S} \right| \), the size of the ground set. In this paper, we show that this algorithm is close to the best possible, namely that there is no o(logn)-approximation algorithm for the test collection problem unless P = NP.

We give approximation algorithms for this problem in the case when all the tests have a small cardinality, significantly improving the performance guarantee achievable by the greedy algorithm. In particular, for instances with test sizes at most k we derive an O(logk) approximation. We show APX-hardness of the version with test sizes at most two, and present an approximation algorithm with ratio \( \tfrac{7} {6} + \varepsilon \) for any fixed ε > 0.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Bjarni V. Halldórsson
    • 1
  • Magnús M. Halldórsson
    • 2
    • 3
  • R. Ravi
    • 4
  1. 1.Dept. of Math. SciencesCarnegie Mellon UniversityUSA
  2. 2.Dept. of Computer ScienceUniversity of IcelandIceland
  3. 3.Iceland Genomics CorpIceland
  4. 4.GSIACarnegie Mellon UniversityUSA

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