35 research outputs found
A Polynomial Delay Algorithm for Enumerating Minimal Dominating Sets in Chordal Graphs
An output-polynomial algorithm for the listing of minimal dominating sets in
graphs is a challenging open problem and is known to be equivalent to the
well-known Transversal problem which asks for an output-polynomial algorithm
for listing the set of minimal hitting sets in hypergraphs. We give a
polynomial delay algorithm to list the set of minimal dominating sets in
chordal graphs, an important and well-studied graph class where such an
algorithm was open for a while.Comment: 13 pages, 1 figure, submitte
Minimal dominating sets enumeration with FPT-delay parameterized by the degeneracy and maximum degree
At STOC 2002, Eiter, Gottlob, and Makino presented a technique called ordered
generation that yields an -delay algorithm listing all minimal
transversals of an -vertex hypergraph of degeneracy . Recently at IWOCA
2019, Conte, Kant\'e, Marino, and Uno asked whether this XP-delay algorithm
parameterized by could be made FPT-delay parameterized by and the
maximum degree , i.e., an algorithm with delay for some computable function . Moreover, as a first step toward
answering that question, they note that the same delay is open for the
intimately related problem of listing all minimal dominating sets in graphs. In
this paper, we answer the latter question in the affirmative.Comment: 18 pages, 2 figure
Perfect Roman Domination and Unique Response Roman Domination
The idea of enumeration algorithms with polynomial delay is to polynomially
bound the running time between any two subsequent solutions output by the
enumeration algorithm. While it is open for more than four decades if all
minimal dominating sets of a graph can be enumerated in output-polynomial time,
it has recently been proven that pointwise-minimal Roman dominating functions
can be enumerated even with polynomial delay. The idea of the enumeration
algorithm was to use polynomial-time solvable extension problems. We use this
as a motivation to prove that also two variants of Roman dominating functions
studied in the literature, named perfect and unique response, can be enumerated
with polynomial delay. This is interesting since Extension Perfect Roman
Domination is W[1]-complete if parameterized by the weight of the given
function and even W[2]-complete if parameterized by the number vertices
assigned 0 in the pre-solution, as we prove. Otherwise, efficient solvability
of extension problems and enumerability with polynomial delay tend to go
hand-in-hand. We achieve our enumeration result by constructing a bijection to
Roman dominating functions, where the corresponding extension problem is
polynomimaltime solvable. Furthermore, we show that Unique Response Roman
Domination is solvable in polynomial time on split graphs, while Perfect Roman
Domination is NP-complete on this graph class, which proves that both
variations, albeit coming with a very similar definition, do differ in some
complexity aspects. This way, we also solve an open problem from the
literature
08431 Abstracts Collection -- Moderately Exponential Time Algorithms
From to , the Dagstuhl Seminar 08431 ``Moderately Exponential Time Algorithms \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available