Parameterized Study of the Test Cover Problem

Crowston, Robert; Gutin, Gregory; Jones, Mark; Saurabh, Saket; Yeo, Anders

doi:10.1007/978-3-642-32589-2_27

Cited by 12 publications

(27 citation statements)

References 16 publications

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“…The marking rules and proofs for (n − k)-Set Cover are relatively simpler when compared to the more complex marking rules and more involved proofs required for (n − k)-Test Cover. These results allow us to generalize, improve and unify several results known in the literature [8,4,5,3].…”

Section: Introductionsupporting

confidence: 74%

“…The following lemmas, that appeared in [5], can be seen to have proofs similar to those of Lemmas 1 and 2.…”

Section: Observation 2 Let (S T ) Be Saturated Let T ∈ S \ T Ands mentioning

confidence: 89%

“…This significantly improves over the running time of the earlier algorithm for the problem [5]. We believe such an approach will be useful for other covering problems as well.…”

Section: Introductionmentioning

confidence: 85%

“…We first describe the Greedy-mini-test(S, k) algorithm from [5]. Like before, this algorithm constructs a collection T of tests from S which has some special properties.…”

Section: Observation 1 For Any Collection Of Tests T and T ∈ T C(tmentioning

confidence: 99%

“…Crowston et al [5] studied, among other parameterizations, the (n − k)-Test Cover problem with the parameter k and gave an algorithm with running time f (k) · (m + n) O(1) for this problem. The function f (k) has a tower of 2's in its exponent.…”

Section: Introductionmentioning

confidence: 99%

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Partially Polynomial Kernels for Set Cover and Test Cover

Basavaraju¹,

Francis²,

Ramanujan³

et al. 2016

SIAM J. Discrete Math.

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In a typical covering problem we are given a universe U of size n, a family S (S could be given implicitly) of size m and an integer k and the objective is to check whether there exists a subfamily S ⊆ S of size at most k satisfying some desired properties. If S is required to contain all the elements of U then it corresponds to the classical Set Cover problem. On the other hand if we require S to satisfy the property that for every pair of elements x, y ∈ U there exists a set S ∈ S such that |S ∩ {x, y}| = 1 then it corresponds to the Test Cover problem. In this paper we consider a natural parameterization of Set Cover and Test Cover. More precisely, we study the (n − k)-Set Cover and (n − k)-Test Cover problems, where the objective is to find a subfamily S of size at most n − k satisfying the respective properties, from the kernelization perspective. It is known in the literature that both (n − k)-Set Cover and (n − k)-Test Cover do not admit polynomial kernels (under some well known complexity theoretic assumptions). However, in this paper we show that they do admit "partially polynomial kernels". More precisely, we give polynomial time algorithms that take as input an instance (U, S, k) of (n − k)-Set Cover ((n − k)-Test Cover) and return an equivalent instance (Ũ,S,k) of (n − k)-Set Cover (respectively (n − k)-Test Cover) withk ≤ k and |Ũ| = O(k 2 ) (|Ũ| = O(k 7 )). These results allow us to generalize, improve and unify several results known in the literature. For example, these immediately imply traditional kernels when input instances satisfy certain "sparsity properties". Using a part of our kernelization algorithm for (n − k)-Set Cover, we also get an improved FPT algorithm for this problem which runs in time O(4 k k O(1) (m + n)) improving over the previous best of O (8 k+o(k) ). On the other hand the partially polynomial kernel for (n − k)-Test Cover implies the first single exponential FPT algorithm, an algorithm with running time O(2 O(k 2 ) (m + n) O(1) ). We believe such an approach will also be useful for other covering problems as well.

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Section: Introductionsupporting

confidence: 74%

“…The following lemmas, that appeared in [5], can be seen to have proofs similar to those of Lemmas 1 and 2.…”

Section: Observation 2 Let (S T ) Be Saturated Let T ∈ S \ T Ands mentioning

confidence: 89%

“…This significantly improves over the running time of the earlier algorithm for the problem [5]. We believe such an approach will be useful for other covering problems as well.…”

Section: Introductionmentioning

confidence: 85%

“…We first describe the Greedy-mini-test(S, k) algorithm from [5]. Like before, this algorithm constructs a collection T of tests from S which has some special properties.…”

Section: Observation 1 For Any Collection Of Tests T and T ∈ T C(tmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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