684 research outputs found
Parameterized Study of the Test Cover Problem
We carry out a systematic study of a natural covering problem, used for
identification across several areas, in the realm of parameterized complexity.
In the {\sc Test Cover} problem we are given a set of items
together with a collection, , of distinct subsets of these items called
tests. We assume that is a test cover, i.e., for each pair of items
there is a test in containing exactly one of these items. The
objective is to find a minimum size subcollection of , which is still a
test cover. The generic parameterized version of {\sc Test Cover} is denoted by
-{\sc Test Cover}. Here, we are given and a
positive integer parameter as input and the objective is to decide whether
there is a test cover of size at most . We study four
parameterizations for {\sc Test Cover} and obtain the following:
(a) -{\sc Test Cover}, and -{\sc Test Cover} are fixed-parameter
tractable (FPT).
(b) -{\sc Test Cover} and -{\sc Test Cover} are
W[1]-hard. Thus, it is unlikely that these problems are FPT
(Non-)existence of Polynomial Kernels for the Test Cover Problem
The input of the Test Cover problem consists of a set of vertices, and a
collection of distinct subsets of , called
tests. A test separates a pair of vertices if A subcollection is a test cover if each
pair of distinct vertices is separated by a test in . The
objective is to find a test cover of minimum cardinality, if one exists. This
problem is NP-hard.
We consider two parameterizations the Test Cover problem with parameter :
(a) decide whether there is a test cover with at most tests, (b) decide
whether there is a test cover with at most tests. Both
parameterizations are known to be fixed-parameter tractable. We prove that none
have a polynomial size kernel unless . 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
Multidimensional Binary Vector Assignment problem: standard, structural and above guarantee parameterizations
In this article we focus on the parameterized complexity of the
Multidimensional Binary Vector Assignment problem (called \BVA). An input of
this problem is defined by disjoint sets , each
composed of binary vectors of size . An output is a set of disjoint
-tuples of vectors, where each -tuple is obtained by picking one vector
from each set . To each -tuple we associate a dimensional vector by
applying the bit-wise AND operation on the vectors of the tuple. The
objective is to minimize the total number of zeros in these vectors. mBVA
can be seen as a variant of multidimensional matching where hyperedges are
implicitly locally encoded via labels attached to vertices, but was originally
introduced in the context of integrated circuit manufacturing.
We provide for this problem FPT algorithms and negative results (-based
results, [2]-hardness and a kernel lower bound) according to several
parameters: the standard parameter i.e. the total number of zeros), as well
as two parameters above some guaranteed values.Comment: 16 pages, 6 figure
Parameterized and Approximation Algorithms for the Load Coloring Problem
Let be two positive integers and let be a graph. The
-Load Coloring Problem (denoted -LCP) asks whether there is a
-coloring such that for every ,
there are at least edges with both endvertices colored . Gutin and Jones
(IPL 2014) studied this problem with . They showed -LCP to be fixed
parameter tractable (FPT) with parameter by obtaining a kernel with at most
vertices. In this paper, we extend the study to any fixed by giving
both a linear-vertex and a linear-edge kernel. In the particular case of ,
we obtain a kernel with less than vertices and less than edges. These
results imply that for any fixed , -LCP is FPT and that the
optimization version of -LCP (where is to be maximized) has an
approximation algorithm with a constant ratio for any fixed
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