Parameterized Study of the Test Cover Problem

  • Robert Crowston
  • Gregory Gutin
  • Mark Jones
  • Saket Saurabh
  • Anders Yeo
Conference paper

DOI: 10.1007/978-3-642-32589-2_27

Part of the Lecture Notes in Computer Science book series (LNCS, volume 7464)
Cite this paper as:
Crowston R., Gutin G., Jones M., Saurabh S., Yeo A. (2012) Parameterized Study of the Test Cover Problem. In: Rovan B., Sassone V., Widmayer P. (eds) Mathematical Foundations of Computer Science 2012. MFCS 2012. Lecture Notes in Computer Science, vol 7464. Springer, Berlin, Heidelberg

Abstract

In this paper we carry out a systematic study of a natural covering problem, used for identification across several areas, in the realm of parameterized complexity. In the Test Cover problem we are given a set [n] = {1,…,n} of items together with a collection, \(\cal T\), of distinct subsets of these items called tests. We assume that \(\cal T\) is a test cover, i.e., for each pair of items there is a test in \(\cal T\) containing exactly one of these items. The objective is to find a minimum size subcollection of \(\cal T\), which is still a test cover. The generic parameterized version of Test Cover is denoted by \(p(k,n,|{\cal T}|)\)-Test Cover. Here, we are given \(([n],\cal{T})\) and a positive integer parameter k as input and the objective is to decide whether there is a test cover of size at most \(p(k,n,|{\cal T}|)\). We study four parameterizations for Test Cover and obtain the following:

(a) k-Test Cover, and (n − k)-Test Cover are fixed-parameter tractable (FPT), i.e., these problems can be solved by algorithms of runtime \(f(k)\cdot poly(n,|{\cal T}|)\), where f(k) is a function of k only.

(b) \((|{\cal T}|-k)\)-Test Cover and (logn + k)-Test Cover are W[1]-hard. Thus, it is unlikely that these problems are FPT.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Robert Crowston
    • 1
  • Gregory Gutin
    • 1
  • Mark Jones
    • 1
  • Saket Saurabh
    • 2
  • Anders Yeo
    • 3
  1. 1.Royal Holloway, University of LondonEghamUK
  2. 2.The Institute of Mathematical SciencesChennaiIndia
  3. 3.University of JohannesburgAuckland ParkSouth Africa

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