1,353 research outputs found
Rainbow domination and related problems on some classes of perfect graphs
Let and let be a graph. A function is a rainbow function if, for every vertex with
, . The rainbow domination number
is the minimum of over all rainbow
functions. We investigate the rainbow domination problem for some classes of
perfect graphs
Total Domishold Graphs: a Generalization of Threshold Graphs, with Connections to Threshold Hypergraphs
A total dominating set in a graph is a set of vertices such that every vertex
of the graph has a neighbor in the set. We introduce and study graphs that
admit non-negative real weights associated to their vertices such that a set of
vertices is a total dominating set if and only if the sum of the corresponding
weights exceeds a certain threshold. We show that these graphs, which we call
total domishold graphs, form a non-hereditary class of graphs properly
containing the classes of threshold graphs and the complements of domishold
graphs, and are closely related to threshold Boolean functions and threshold
hypergraphs. We present a polynomial time recognition algorithm of total
domishold graphs, and characterize graphs in which the above property holds in
a hereditary sense. Our characterization is obtained by studying a new family
of hypergraphs, defined similarly as the Sperner hypergraphs, which may be of
independent interest.Comment: 19 pages, 1 figur
Bidimensionality and EPTAS
Bidimensionality theory is a powerful framework for the development of
metaalgorithmic techniques. It was introduced by Demaine et al. as a tool to
obtain sub-exponential time parameterized algorithms for problems on H-minor
free graphs. Demaine and Hajiaghayi extended the theory to obtain PTASs for
bidimensional problems, and subsequently improved these results to EPTASs.
Fomin et. al related the theory to the existence of linear kernels for
parameterized problems. In this paper we revisit bidimensionality theory from
the perspective of approximation algorithms and redesign the framework for
obtaining EPTASs to be more powerful, easier to apply and easier to understand.
Two of the most widely used approaches to obtain PTASs on planar graphs are
the Lipton-Tarjan separator based approach, and Baker's approach. Demaine and
Hajiaghayi strengthened both approaches using bidimensionality and obtained
EPTASs for a multitude of problems. We unify the two strenghtened approaches to
combine the best of both worlds. At the heart of our framework is a
decomposition lemma which states that for "most" bidimensional problems, there
is a polynomial time algorithm which given an H-minor-free graph G as input and
an e > 0 outputs a vertex set X of size e * OPT such that the treewidth of G n
X is f(e). Here, OPT is the objective function value of the problem in question
and f is a function depending only on e. This allows us to obtain EPTASs on
(apex)-minor-free graphs for all problems covered by the previous framework, as
well as for a wide range of packing problems, partial covering problems and
problems that are neither closed under taking minors, nor contractions. To the
best of our knowledge for many of these problems including cycle packing,
vertex-h-packing, maximum leaf spanning tree, and partial r-dominating set no
EPTASs on planar graphs were previously known
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