4,693 research outputs found
On parallel versus sequential approximation
In this paper we deal with the class NCX of NP Optimization problems that are approximable within constant ratio in NC. This class is the parallel counterpart of the class APX. Our main motivation here is to reduce the study of sequential and parallel approximability to the same framework. To this aim, we first introduce a new kind of NC-reduction that preserves the relative error of the approximate solutions and show that the class NCX has {em complete} problems under this reducibility.
An important subset of NCX is the class MAXSNP, we show that MAXSNP-complete problems have a threshold on the parallel approximation ratio that is, there are positive constants , such that although the problem can be approximated in P within it cannot be approximated in NC within epsilon_2$, unless P=NC. This result is attained by showing that the problem of approximating the value obtained through a non-oblivious local search algorithm is P-complete, for some values of the approximation ratio. Finally, we show that approximating through non-oblivious local search is in average NC.Postprint (published version
On the Complexity of Making a Distinguished Vertex Minimum or Maximum Degree by Vertex Deletion
In this paper, we investigate the approximability of two node deletion
problems. Given a vertex weighted graph and a specified, or
"distinguished" vertex , MDD(min) is the problem of finding a minimum
weight vertex set such that becomes the
minimum degree vertex in ; and MDD(max) is the problem of
finding a minimum weight vertex set such that
becomes the maximum degree vertex in . These are known
-complete problems and have been studied from the parameterized complexity
point of view in previous work. Here, we prove that for any ,
both the problems cannot be approximated within a factor , unless . We also show that for any
, MDD(min) cannot be approximated within a factor on bipartite graphs, unless , and that for any , MDD(max) cannot be approximated within a
factor on bipartite graphs, unless . We give an factor approximation algorithm
for MDD(max) on general graphs, provided the degree of is . We
then show that if the degree of is , a similar result holds
for MDD(min). We prove that MDD(max) is -complete on 3-regular unweighted
graphs and provide an approximation algorithm with ratio when is a
3-regular unweighted graph. In addition, we show that MDD(min) can be solved in
polynomial time when is a regular graph of constant degree.Comment: 16 pages, 4 figures, submitted to Elsevier's Journal of Discrete
Algorithm
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
On the Size and the Approximability of Minimum Temporally Connected Subgraphs
We consider temporal graphs with discrete time labels and investigate the
size and the approximability of minimum temporally connected spanning
subgraphs. We present a family of minimally connected temporal graphs with
vertices and edges, thus resolving an open question of (Kempe,
Kleinberg, Kumar, JCSS 64, 2002) about the existence of sparse temporal
connectivity certificates. Next, we consider the problem of computing a minimum
weight subset of temporal edges that preserve connectivity of a given temporal
graph either from a given vertex r (r-MTC problem) or among all vertex pairs
(MTC problem). We show that the approximability of r-MTC is closely related to
the approximability of Directed Steiner Tree and that r-MTC can be solved in
polynomial time if the underlying graph has bounded treewidth. We also show
that the best approximation ratio for MTC is at least and at most , for
any constant , where is the number of temporal edges and
is the maximum degree of the underlying graph. Furthermore, we prove
that the unweighted version of MTC is APX-hard and that MTC is efficiently
solvable in trees and -approximable in cycles
Inapproximability of Combinatorial Optimization Problems
We survey results on the hardness of approximating combinatorial optimization
problems
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