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Parameterizations of Test Cover with Bounded Test Sizes

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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 \(\frac{2(n-1)}{r+1}\).

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Notes

  1. Notice that, in the literature, the vertices are also called items and the edges tests. We provide basic terminology and notation on hypergraphs in Sect. 2.

  2. We provide basic notions on parameterized algorithmics in the end of this section.

  3. If \(n'\) is not divisible by \(r-1\), add \(r-1-n'\)mod\((r-1)\) new vertices to each of the \(X_i\)’s and the same number of new pairwise disjoint edges which together contain each of the new vertices exactly once. Observe that this produces an instance which is equivalent to the original one.

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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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Authors and Affiliations

  1. Royal Holloway, University of London, Egham, Surrey, TW20 0EX, UK

    R. Crowston, G. Gutin, M. Jones & G. Muciaccia

  2. Singapore University of Technology and Design, 20 Dover Drive, Singapore, 138682, Singapore

    A. Yeo

  3. University of Johannesburg, Auckland Park, 2006, South Africa

    A. Yeo

Authors
  1. R. Crowston
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  2. G. Gutin
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  3. M. Jones
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  4. G. Muciaccia
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  5. A. Yeo
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Correspondence to G. Gutin.

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Crowston, R., Gutin, G., Jones, M. et al. Parameterizations of Test Cover with Bounded Test Sizes. Algorithmica 74, 367–384 (2016). https://doi.org/10.1007/s00453-014-9948-7

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