Abstract.
In the test cover problem a set of m items is given together with a collection of subsets, called tests. A smallest subcollection of tests is to be selected such that for each pair of items there is a test in the selection that contains exactly one of the two items. It is known that the problem is NP-hard and that the greedy algorithm has a performance ratio O(log m). We observe that, unless P=NP, no polynomial-time algorithm can do essentially better. For the case that each test contains at most k items, we give an O(log k)-approximation algorithm. We pay special attention to the case that each test contains at most two items. A strong relation with a problem of packing paths in a graph is established, which implies that even this special case is NP-hard. We prove APX-hardness of both problems, derive performance guarantees for greedy algorithms, and discuss the performance of a series of local improvement heuristics.
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Partially supported by the Future and Emerging Technologies Programme of the EU under contract number IST-1999-14186 (ALCOM-FT).
Partially supported by a Merck Computational Biology and Chemistry Program Graduate Fellowship from the Merck Company Foundation.
Also Iceland Genomics Corporation
Partially supported by subcontract No. 16082-RFP-00-2C in the area of ``Combinatorial Optimization in Biology (XAXE),'' Los Alamos National Laboratories, and NSF grant CCR-0105548.
Mathematics Subject Classification: 90B27
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De Bontridder, K., Halldórsson, B., Halldórsson, M. et al. Approximation algorithms for the test cover problem. Math. Program., Ser. B 98, 477–491 (2003). https://doi.org/10.1007/s10107-003-0414-6
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DOI: https://doi.org/10.1007/s10107-003-0414-6