1,676 research outputs found
Polynomial Kernels for Generalized Domination Problems
In this paper, we study the parameterized complexity of a generalized
domination problem called the [] Dominating Set problem. This
problem generalizes a large number of problems including the Minimum Dominating
Set problem and its many variants. The parameterized complexity of the
[] Dominating Set problem parameterized by treewidth is well
studied. Here the properties of the sets and that make the
problem tractable are identified [1]. We consider a larger parameter and
investigate the existence of polynomial sized kernels. When and
are finite, we identify the exact condition when the [] Dominating Set problem parameterized by vertex cover admits polynomial
kernels. Our lower and upper bound results can also be extended to more general
conditions and provably smaller parameters as well.Comment: 19 pages, 6 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
(Total) Vector Domination for Graphs with Bounded Branchwidth
Given a graph of order and an -dimensional non-negative
vector , called demand vector, the vector domination
(resp., total vector domination) is the problem of finding a minimum
such that every vertex in (resp., in ) has
at least neighbors in . The (total) vector domination is a
generalization of many dominating set type problems, e.g., the dominating set
problem, the -tuple dominating set problem (this is different from the
solution size), and so on, and its approximability and inapproximability have
been studied under this general framework. In this paper, we show that a
(total) vector domination of graphs with bounded branchwidth can be solved in
polynomial time. This implies that the problem is polynomially solvable also
for graphs with bounded treewidth. Consequently, the (total) vector domination
problem for a planar graph is subexponential fixed-parameter tractable with
respectto , where is the size of solution.Comment: 16 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
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