# On the Approximability of the Minimum Test Collection Problem

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## Abstract

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*(log*n*)-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*(log*k*) 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.

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### References

- 1.M. Abbink.
*The test cover problem*. Master’s thesis, University of Amsterdam, 1995.Google Scholar - 2.S. Arora and M. Sudan. Improved low degree testing and its applications. In
*Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing*, pages 485–495, El Paso, Texas, 4–6 May 1997.Google Scholar - 3.K.M. J. De Bontridder. Methods for solving the test cover problem. Master’s thesis, Eindhoven University of Technology, 1997.Google Scholar
- 4.U. Feige. A threshold of ln
*n*for approximating set cover.*Journal of the ACM*, 45:634–652, 1998.MATHCrossRefMathSciNetGoogle Scholar - 5.U. Feige and J. Killian. Zero knowledge and the chromatic number.
*J. Comput. System Sci.*, 57:187–199, 1998.MATHCrossRefMathSciNetGoogle Scholar - 6.M.R. Garey and D.S. Johnson.
*Computers and Intractability: A Guide to the Theory of NP-completeness*. W.H. Freeman and Company, 1979.Google Scholar - 7.B.V. Halld’orsson, J.S. Minden, and R. Ravi. PIER: Protein identification by epitope recognition. In N. El-Mabrouk, T. Lengauer, and D. Sankoff, editors,
*Currents in Computational Molecular Biology 2001*, pages 109–110, 2001.Google Scholar - 8.M.M. Halldórsson. A still better performance guarentee for approximate graph coloring.
*Information Processing Letters*, 45:19–23, 1993.MATHCrossRefMathSciNetGoogle Scholar - 9.D. S. Hochbaum
*Approximation Algorithms for NP-hard problems*. PWS, 1997.Google Scholar - 10.C.A.J. Hurkens and A. Schrijver. On the size of systems of sets every
*t*of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems.*SIAM J. Discrete Mathematics*, 2:68–72, 1989.MATHCrossRefMathSciNetGoogle Scholar - 11.D.S. Johnson. Approximation algorithms for combinatorial problems.
*J. Comput. System Sci*, 9:256–278, 1972.CrossRefGoogle Scholar - 12.V. Kann Maximum bounded 3-dimensional matching is MAX SNP-complete.
*Information Processing Letters*, 37:27–35, 1991.MATHCrossRefMathSciNetGoogle Scholar - 13.R. Loulou. Minimal cut cover of a graph with an application to the testing of electronic boards.
*Operations Research Letters*, 12:301–305, 1992.MATHCrossRefMathSciNetGoogle Scholar - 14.L. Lovász. On the ratio of optimal integral and fractional covers.
*Discrete Mathematics*, 13:383–390, 1975.MATHCrossRefMathSciNetGoogle Scholar - 15.C. Lund and M. Yannakakis. On the hardness of approximating minimization problems.
*Journal of the ACM*, 41(5):960–981, 1994.MATHCrossRefMathSciNetGoogle Scholar - 16.B.M.E. Moret and H.D. Shapiro. On minimizing a set of tests.
*SIAM Journal on Scientific and Statistical Computing*, 6:983–1003, 1985.CrossRefGoogle Scholar - 17.R. Motwani and J. Naor. On exact and approximate cut covers of graphs. Technical report, Stanford University, 1994.Google Scholar