120 research outputs found
Parameterized (in)approximability of subset problems
We discuss approximability and inapproximability in FPT-time for a large
class of subset problems where a feasible solution is a subset of the input
data and the value of is . The class handled encompasses many
well-known graph, set, or satisfiability problems such as Dominating Set,
Vertex Cover, Set Cover, Independent Set, Feedback Vertex Set, etc. In a first
time, we introduce the notion of intersective approximability that generalizes
the one of safe approximability and show strong parameterized inapproximability
results for many of the subset problems handled. Then, we study approximability
of these problems with respect to the dual parameter where is the
size of the instance and the standard parameter. More precisely, we show
that under such a parameterization, many of these problems, while
W[]-hard, admit parameterized approximation schemata.Comment: 7 page
Parametrized Complexity of Weak Odd Domination Problems
Given a graph , a subset of vertices is a weak odd
dominated (WOD) set if there exists such that
every vertex in has an odd number of neighbours in . denotes
the size of the largest WOD set, and the size of the smallest
non-WOD set. The maximum of and , denoted
, plays a crucial role in quantum cryptography. In particular
deciding, given a graph and , whether is of
practical interest in the design of graph-based quantum secret sharing schemes.
The decision problems associated with the quantities , and
are known to be NP-Complete. In this paper, we consider the
approximation of these quantities and the parameterized complexity of the
corresponding problems. We mainly prove the fixed-parameter intractability
(W-hardness) of these problems. Regarding the approximation, we show that
, and admit a constant factor approximation
algorithm, and that and have no polynomial approximation
scheme unless P=NP.Comment: 16 pages, 5 figure
The Parameterized Complexity of Domination-type Problems and Application to Linear Codes
We study the parameterized complexity of domination-type problems.
(sigma,rho)-domination is a general and unifying framework introduced by Telle:
a set D of vertices of a graph G is (sigma,rho)-dominating if for any v in D,
|N(v)\cap D| in sigma and for any $v\notin D, |N(v)\cap D| in rho. We mainly
show that for any sigma and rho the problem of (sigma,rho)-domination is W[2]
when parameterized by the size of the dominating set. This general statement is
optimal in the sense that several particular instances of
(sigma,rho)-domination are W[2]-complete (e.g. Dominating Set). We also prove
that (sigma,rho)-domination is W[2] for the dual parameterization, i.e. when
parameterized by the size of the dominated set. We extend this result to a
class of domination-type problems which do not fall into the
(sigma,rho)-domination framework, including Connected Dominating Set. We also
consider problems of coding theory which are related to domination-type
problems with parity constraints. In particular, we prove that the problem of
the minimal distance of a linear code over Fq is W[2] for both standard and
dual parameterizations, and W[1]-hard for the dual parameterization.
To prove W[2]-membership of the domination-type problems we extend the
Turing-way to parameterized complexity by introducing a new kind of non
deterministic Turing machine with the ability to perform `blind' transitions,
i.e. transitions which do not depend on the content of the tapes. We prove that
the corresponding problem Short Blind Multi-Tape Non-Deterministic Turing
Machine is W[2]-complete. We believe that this new machine can be used to prove
W[2]-membership of other problems, not necessarily related to dominationComment: 19 pages, 2 figure
A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms
Parameterization and approximation are two popular ways of coping with
NP-hard problems. More recently, the two have also been combined to derive many
interesting results. We survey developments in the area both from the
algorithmic and hardness perspectives, with emphasis on new techniques and
potential future research directions
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
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