Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
On the Approximability of the Minimum Test Collection Problem
ESA '01 Proceedings of the 9th Annual European Symposium on Algorithms
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
On problems without polynomial kernels
Journal of Computer and System Sciences
Parametric duality and kernelization: lower bounds and upper bounds on kernel size
STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science
Parameterized study of the test cover problem
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
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The input of the Test Cover problem consists of a set V of vertices, and a collection E={E"1,...,E"m} of distinct subsets of V, called tests. A test E"q separates a pair v"i,v"j of vertices if |{v"i,v"j}@?E"q|=1. A subcollection T@?E is a test cover if each pair v"i,v"j of distinct vertices is separated by a test in T. The objective is to find a test cover of minimum cardinality, if one exists. This problem is NP-hard. We consider two parameterizations of the Test Cover problem with parameter k: (a) decide whether there is a test cover with at most k tests, (b) decide whether there is a test cover with at most |V|-k tests. Both parameterizations are known to be fixed-parameter tractable. We prove that none have a polynomial size kernel unless NP@?coNP/poly. Our proofs use the cross-composition method recently introduced by Bodlaender et al. (2011) and parametric duality introduced by Chen et al. (2005). The result for the parameterization (a) was an open problem (private communications with Henning Fernau and Jiong Guo, Jan.-Feb. 2012). We also show that the parameterization (a) admits a polynomial size kernel if the size of each test is upper-bounded by a constant.