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Algorithmica

, Volume 74, Issue 1, pp 367–384 | Cite as

Parameterizations of Test Cover with Bounded Test Sizes

  • R. Crowston
  • G. Gutin
  • M. Jones
  • G. Muciaccia
  • A. Yeo
Article

Abstract

In the Test Cover problem we are given a hypergraph \(H=(V, {\mathcal {E}})\) with \(|V|=n, |{\mathcal {E}}|=m\), and we assume that \({\mathcal {E}}\) is a test cover, i.e. for every pair of vertices \(x_i, x_j\), there exists an edge \(e \in {\mathcal {E}}\) such that \(|\{x_i,x_j\}\cap e|=1\). The objective is to find a minimum subset of \({\mathcal {E}}\) which is a test cover. The problem is used for identification across many areas, and is NP-complete. From a parameterized complexity standpoint, many natural parameterizations of Test Cover are either \(W[1]\)-complete or have no polynomial kernel unless \(coNP\subseteq NP/poly\), and thus are unlikely to be solveable efficiently. However, in practice the size of the edges is often bounded. In this paper we study the parameterized complexity of Test-\(r\)-Cover, the restriction of Test Cover in which each edge contains at most \(r \ge 2\) vertices. In contrast to the unbounded case, we show that the following below-bound parameterizations of Test-\(r\)-Cover are fixed-parameter tractable with a polynomial kernel: (1) Decide whether there exists a test cover of size \(n-k\), and (2) decide whether there exists a test cover of size \(m-k\), where \(k\) is the parameter. In addition, we prove a new lower bound \(\lceil \frac{2(n-1)}{r+1} \rceil \) on the minimum size of a test cover when the size of each edge is bounded by \(r\). Test-\(r\)-Cover parameterized above this bound is unlikely to be fixed-parameter tractable; in fact, we show that it is para-NP-complete, as it is NP-hard to decide whether an instance of Test-\(r\)-Cover has a test cover of size exactly \(\frac{2(n-1)}{r+1}\).

Keywords

Test cover Tests of bounded sizes Fixed-parameter tractability Polynomial kernel 

Notes

Acknowledgments

We are very grateful to one of the referees for numerous suggestions which allowed us to improve the presentation. We are also grateful to Manu Basavaraju and Mathew Francis for carefully reading an earlier version of this paper and informing us of a subtle flaw in Theorem 9 which led us to changing the proof substantially.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • R. Crowston
    • 1
  • G. Gutin
    • 1
  • M. Jones
    • 1
  • G. Muciaccia
    • 1
  • A. Yeo
    • 2
    • 3
  1. 1.Royal Holloway, University of LondonEghamUK
  2. 2.Singapore University of Technology and DesignSingaporeSingapore
  3. 3.University of JohannesburgAuckland ParkSouth Africa

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