80 research outputs found
Parameterized algorithms of fundamental NP-hard problems: a survey
Parameterized computation theory has developed rapidly over the last two decades. In theoretical computer science, it has attracted considerable attention for its theoretical value and significant guidance in many practical applications. We give an overview on parameterized algorithms for some fundamental NP-hard problems, including MaxSAT, Maximum Internal Spanning Trees, Maximum Internal Out-Branching, Planar (Connected) Dominating Set, Feedback Vertex Set, Hyperplane Cover, Vertex Cover, Packing and Matching problems. All of these problems have been widely applied in various areas, such as Internet of Things, Wireless Sensor Networks, Artificial Intelligence, Bioinformatics, Big Data, and so on. In this paper, we are focused on the algorithms’ main idea and algorithmic techniques, and omit the details of them
Markov-Chain-Based Heuristics for the Feedback Vertex Set Problem for Digraphs
A feedback vertex set (FVS) of an undirected or directed graph G=(V, A) is a set F such that G-F is acyclic. The minimum feedback vertex set problem asks for a FVS of G of minimum cardinality whereas the weighted minimum feedback vertex set problem consists of determining a FVS F of minimum weight w(F) given a real-valued weight function w. Both problems are NP-hard [Karp72]. Nethertheless, they have been found to have applications in many fields. So one is naturally interested in approximation algorithms. While most of the existing approximation algorithms for feedback vertex set problems rely on local properties of G only, this thesis explores strategies that use global information about G in order to determine good solutions. The pioneering work in this direction has been initiated by Speckenmeyer [Speckenmeyer89]. He demonstrated the use of Markov chains for determining low cardinality FVSs. Based on his ideas, new approximation algorithms are developed for both the unweighted and the weighted minimum feedback vertex set problem for digraphs. According to the experimental results presented in this thesis, these new algorithms outperform all other existing approximation algorithms. An additional contribution, not related to Markov chains, is the identification of a new class of digraphs G=(V, A) which permit the determination of an optimum FVS in time O(|V|^4). This class strictly encompasses the completely contractible graphs [Levy/Low88]
Exact Localisations of Feedback Sets
The feedback arc (vertex) set problem, shortened FASP (FVSP), is to transform
a given multi digraph into an acyclic graph by deleting as few arcs
(vertices) as possible. Due to the results of Richard M. Karp in 1972 it is one
of the classic NP-complete problems. An important contribution of this paper is
that the subgraphs , of all elementary
cycles or simple cycles running through some arc , can be computed in
and , respectively. We use
this fact and introduce the notion of the essential minor and isolated cycles,
which yield a priori problem size reductions and in the special case of so
called resolvable graphs an exact solution in . We show
that weighted versions of the FASP and FVSP possess a Bellman decomposition,
which yields exact solutions using a dynamic programming technique in times
and
, where , , respectively. The parameters can
be computed in , ,
respectively and denote the maximal dimension of the cycle space of all
appearing meta graphs, decoding the intersection behavior of the cycles.
Consequently, equal zero if all meta graphs are trees. Moreover, we
deliver several heuristics and discuss how to control their variation from the
optimum. Summarizing, the presented results allow us to suggest a strategy for
an implementation of a fast and accurate FASP/FVSP-SOLVER
Cycle killer... qu'est-ce que c'est? On the comparative approximability of hybridization number and directed feedback vertex set
We show that the problem of computing the hybridization number of two rooted
binary phylogenetic trees on the same set of taxa X has a constant factor
polynomial-time approximation if and only if the problem of computing a
minimum-size feedback vertex set in a directed graph (DFVS) has a constant
factor polynomial-time approximation. The latter problem, which asks for a
minimum number of vertices to be removed from a directed graph to transform it
into a directed acyclic graph, is one of the problems in Karp's seminal 1972
list of 21 NP-complete problems. However, despite considerable attention from
the combinatorial optimization community it remains to this day unknown whether
a constant factor polynomial-time approximation exists for DFVS. Our result
thus places the (in)approximability of hybridization number in a much broader
complexity context, and as a consequence we obtain that hybridization number
inherits inapproximability results from the problem Vertex Cover. On the
positive side, we use results from the DFVS literature to give an O(log r log
log r) approximation for hybridization number, where r is the value of an
optimal solution to the hybridization number problem
Algorithm design techniques for parameterized graph modification problems
Diese Arbeit beschaeftigt sich mit dem Entwurf parametrisierter Algorithmen fuer Graphmodifikationsprobleme wie Feedback Vertex Set, Multicut in Trees, Cluster Editing und Closest 3-Leaf Powers. Anbei wird die Anwendbarkeit von vier Technicken zur Entwicklung parametrisierter Algorithmen, naemlich, Datenreduktion, Suchbaum, Iterative Kompression und Dynamische Programmierung, fuer solche Graphmodifikationsprobleme untersucht
Kernelization for Counting Problems on Graphs: Preserving the Number of Minimum Solutions
A kernelization for a parameterized decision problem is a
polynomial-time preprocessing algorithm that reduces any parameterized instance
into an instance whose size is bounded by a function of
alone and which has the same yes/no answer for . Such
preprocessing algorithms cannot exist in the context of counting problems, when
the answer to be preserved is the number of solutions, since this number can be
arbitrarily large compared to . However, we show that for counting minimum
feedback vertex sets of size at most , and for counting minimum dominating
sets of size at most in a planar graph, there is a polynomial-time
algorithm that either outputs the answer or reduces to an instance of
size polynomial in with the same number of minimum solutions. This shows
that a meaningful theory of kernelization for counting problems is possible and
opens the door for future developments. Our algorithms exploit that if the
number of solutions exceeds , the size of the input is
exponential in terms of so that the running time of a parameterized
counting algorithm can be bounded by . Otherwise, we can use
gadgets that slightly increase to represent choices among
options by only vertices.Comment: Extended abstract appears in the proceedings of IPEC 202
Finding secluded places of special interest in graphs.
Finding a vertex subset in a graph that satisfies a certain property is one of the most-studied topics
in algorithmic graph theory. The focus herein is often on minimizing or maximizing the size
of the solution, that is, the size of the desired vertex set. In several applications, however, we also
want to limit the “exposure” of the solution to the rest of the graph. This is the case, for example,
when the solution represents persons that ought to deal with sensitive information or a segregated
community. In this work, we thus explore the (parameterized) complexity of finding such secluded
vertex subsets for a wide variety of properties that they shall fulfill. More precisely, we study the
constraint that the (open or closed) neighborhood of the solution shall be bounded by a parameter
and the influence of this constraint on the complexity of minimizing separators, feedback vertex
sets, F-free vertex deletion sets, dominating sets, and the maximization of independent sets
Finding secluded places of special interest in graphs
Finding a vertex subset in a graph that satisfies a certain property is one of the most-studied topics in algorithmic graph theory. The focus herein is often on minimizing or maximizing the size of the solution, that is, the size of the desired vertex set. In several applications, however, we also want to limit the “exposure” of the solution to the rest of the graph. This is the case, for example, when the solution represents persons that ought to deal with sensitive information or a segregated community. In this work, we thus explore the (parameterized) complexity of finding such secluded vertex subsets for a wide variety of properties that they shall fulfill. More precisely, we study the constraint that the (open or closed) neighborhood of the solution shall be bounded by a parameter and the influence of this constraint on the complexity of minimizing separators, feedback vertex sets, F-free vertex deletion sets, dominating sets, and the maximization of independent sets
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