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)


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.


Approximation Algorithm Greedy Algorithm Complete Graph Test Size Performance Guarantee 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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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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