12,185 research outputs found
A distributed algorithm to find k-dominating sets
We consider a connected undirected graph with nodes and
edges. A -dominating set in is a set of nodes having the property
that every node in is at most edges away from at least one node in .
Finding a -dominating set of minimum size is NP-hard. We give a new
synchronous distributed algorithm to find a -dominating set in of size
no greater than . Our algorithm requires
time and messages to run. It has the same time
complexity as the best currently known algorithm, but improves on that
algorithm's message complexity and is, in addition, conceptually simpler.Comment: To appear in Discrete Applied Mathematic
Isomorphism of graph classes related to the circular-ones property
We give a linear-time algorithm that checks for isomorphism between two 0-1
matrices that obey the circular-ones property. This algorithm leads to
linear-time isomorphism algorithms for related graph classes, including Helly
circular-arc graphs, \Gamma-circular-arc graphs, proper circular-arc graphs and
convex-round graphs.Comment: 25 pages, 9 figure
Near-Optimal Distributed Approximation of Minimum-Weight Connected Dominating Set
This paper presents a near-optimal distributed approximation algorithm for
the minimum-weight connected dominating set (MCDS) problem. The presented
algorithm finds an approximation in rounds,
where is the network diameter and is the number of nodes.
MCDS is a classical NP-hard problem and the achieved approximation factor
is known to be optimal up to a constant factor, unless P=NP.
Furthermore, the round complexity is known to be
optimal modulo logarithmic factors (for any approximation), following [Das
Sarma et al.---STOC'11].Comment: An extended abstract version of this result appears in the
proceedings of 41st International Colloquium on Automata, Languages, and
Programming (ICALP 2014
Parameterized and approximation complexity of the detection pair problem in graphs
We study the complexity of the problem DETECTION PAIR. A detection pair of a
graph is a pair of sets of detectors with , the
watchers, and , the listeners, such that for every pair
of vertices that are not dominated by a watcher of , there is a listener of
whose distances to and to are different. The goal is to minimize
. This problem generalizes the two classic problems DOMINATING SET and
METRIC DIMENSION, that correspond to the restrictions and
, respectively. DETECTION PAIR was recently introduced by Finbow,
Hartnell and Young [A. S. Finbow, B. L. Hartnell and J. R. Young. The
complexity of monitoring a network with both watchers and listeners.
Manuscript, 2015], who proved it to be NP-complete on trees, a surprising
result given that both DOMINATING SET and METRIC DIMENSION are known to be
linear-time solvable on trees. It follows from an existing reduction by Hartung
and Nichterlein for METRIC DIMENSION that even on bipartite subcubic graphs of
arbitrarily large girth, DETECTION PAIR is NP-hard to approximate within a
sub-logarithmic factor and W[2]-hard (when parameterized by solution size). We
show, using a reduction to SET COVER, that DETECTION PAIR is approximable
within a factor logarithmic in the number of vertices of the input graph. Our
two main results are a linear-time -approximation algorithm and an FPT
algorithm for DETECTION PAIR on trees.Comment: 13 page
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