Parameterized Study of the Test Cover Problem

  • Robert Crowston
  • Gregory Gutin
  • Mark Jones
  • Saket Saurabh
  • Anders Yeo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7464)

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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