2,627 research outputs found
On the algorithmic complexity of twelve covering and independence parameters of graphs
The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs
Complexity and algorithms related to two classes of graph problems
This thesis addresses the problems associated with conversions on graphs and editing by removing a matching. We study the f-reversible processes, which are those associated with a threshold value for each vertex, and whose dynamics depends on the number of neighbors with different state for each vertex. We set a tight upper bound for the period and transient lengths, characterize all trees that reach the maximum transient length for 2-reversible processes, and we show that determining the size of a minimum conversion set is NP-hard. We show that the AND-OR model defines a convexity on graphs. We show results of NP-completeness and efficient algorithms for certain convexity parameters for this new one, as well as approximate algorithms. We introduce the concept of generalized threshold processes, where the results are NP-completeness and efficient algorithms for both non relaxed and relaxed versions. We study the problem of deciding whether a given graph admits a removal of a matching in order to destroy all cycles. We show that this problem is NP-hard even for subcubic graphs, but admits efficient solution for several graph classes. We study the problem of deciding whether a given graph admits a removal of a matching in order to destroy all odd cycles. We show that this problem is NP-hard even for planar graphs with bounded degree, but admits efficient solution for some graph classes. We also show parameterized results.Esta tese aborda problemas associados a conversões em grafos e de edição pela remoção de um emparelhamento. Estudamos processos f-reversÃveis, que são aqueles associados a um valor de limiar para cada vértice e cuja dinâmica depende da quantidade de vizinhos com estado contrário para cada vértice. Estabelecemos um limite superior justo para o tamanho do perÃodo e transiente, caracterizamos todas as árvores que alcançam o transiente máximo em processos 2-reversÃveis e mostramos que determinar o tamanho de um conjunto conversor mÃnimo é NP-difÃcil. Mostramos que o modelo AND-OR define uma convexidade sobre grafos. Mostramos resultados de NP-completude e algoritmos eficientes para certos parâmetros de convexidade para esta nova, assim como algoritmos aproximativos. Introduzimos o conceito de processos de limiar generalizados, onde mostramos resultados de NP-completude e algoritmos eficientes para ambas as versões não relaxada e relaxada. Estudamos o problema de decidir se um dado grafo admite uma remoção de um emparelhamento de modo a remover todos os ciclos. Mostramos que este problema é NP-difÃcil mesmo para grafos subcúbicos, mas admite solução eficiente para várias classes de grafos. Estudamos o problema de decidir se um dado grafo admite uma remoção de um emparelhamento de modo a remover todos os ciclos Ãmpares. Mostramos que este problema é NP-difÃcil mesmo para grafos planares com grau limitado, mas admite solução eficiente para algumas classes de grafos. Mostramos também resultados parametrizados
Solving Hard Graph Problems with Combinatorial Computing and Optimization
Many problems arising in graph theory are difficult by nature, and finding solutions to large or complex instances of them often require the use of computers. As some such problems are -hard or lie even higher in the polynomial hierarchy, it is unlikely that efficient, exact algorithms will solve them. Therefore, alternative computational methods are used. Combinatorial computing is a branch of mathematics and computer science concerned with these methods, where algorithms are developed to generate and search through combinatorial structures in order to determine certain properties of them. In this thesis, we explore a number of such techniques, in the hopes of solving specific problem instances of interest.
Three separate problems are considered, each of which is attacked with different methods of combinatorial computing and optimization. The first, originally proposed by ErdH{o}s and Hajnal in 1967, asks to find the Folkman number , defined as the smallest order of a -free graph that is not the union of two triangle-free graphs. A notoriously difficult problem associated with Ramsey theory, the best known bounds on it prior to this work were . We improve the upper bound to using a combination of known methods and the Goemans-Williamson semi-definite programming relaxation of MAX-CUT. The second problem of interest is the Ramsey number , which is the smallest such that any -vertex graph contains a cycle of length four or an independent set of order . With the help of combinatorial algorithms, we determine and using large-scale computations on the Open Science Grid. Finally, we explore applications of the well-known Lenstra-Lenstra-Lov\u27{a}sz (LLL) algorithm, a polynomial-time algorithm that, when given a basis of a lattice, returns a basis for the same lattice with relatively short vectors. The main result of this work is an application to graph domination, where certain hard instances are solved using this algorithm as a heuristic
New Algorithms for Mixed Dominating Set
A mixed dominating set is a collection of vertices and edges that dominates
all vertices and edges of a graph. We study the complexity of exact and
parameterized algorithms for \textsc{Mixed Dominating Set}, resolving some open
questions. In particular, we settle the problem's complexity parameterized by
treewidth and pathwidth by giving an algorithm running in time
(improving the current best ), as well as a lower bound showing
that our algorithm cannot be improved under the Strong Exponential Time
Hypothesis (SETH), even if parameterized by pathwidth (improving a lower bound
of ). Furthermore, by using a simple but so far
overlooked observation on the structure of minimal solutions, we obtain
branching algorithms which improve both the best known FPT algorithm for this
problem, from to , and the best known
exponential-time exact algorithm, from and exponential space, to
and polynomial space.Comment: This paper has been accepted to IPEC 202
On vertex independence number of uniform hypergraphs
Abstract
Let H be an r-uniform hypergraph with r ≥ 2 and let α(H) be its vertex independence number. In the paper bounds of α(H) are given for different uniform hypergraphs: if H has no isolated vertex, then in terms of the degrees, and for triangle-free linear H in terms of the order and average degree.</jats:p
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