90 research outputs found
Graph classes and forbidden patterns on three vertices
This paper deals with graph classes characterization and recognition. A
popular way to characterize a graph class is to list a minimal set of forbidden
induced subgraphs. Unfortunately this strategy usually does not lead to an
efficient recognition algorithm. On the other hand, many graph classes can be
efficiently recognized by techniques based on some interesting orderings of the
nodes, such as the ones given by traversals.
We study specifically graph classes that have an ordering avoiding some
ordered structures. More precisely, we consider what we call patterns on three
nodes, and the recognition complexity of the associated classes. In this
domain, there are two key previous works. Damashke started the study of the
classes defined by forbidden patterns, a set that contains interval, chordal
and bipartite graphs among others. On the algorithmic side, Hell, Mohar and
Rafiey proved that any class defined by a set of forbidden patterns can be
recognized in polynomial time. We improve on these two works, by characterizing
systematically all the classes defined sets of forbidden patterns (on three
nodes), and proving that among the 23 different classes (up to complementation)
that we find, 21 can actually be recognized in linear time.
Beyond this result, we consider that this type of characterization is very
useful, leads to a rich structure of classes, and generates a lot of open
questions worth investigating.Comment: Third version version. 38 page
Perfect Graphs
This chapter is a survey on perfect graphs with an algorithmic flavor. Our emphasis is on important classes of perfect graphs for which there are fast and efficient recognition and optimization algorithms. The classes of graphs we discuss in this chapter are chordal, comparability, interval, perfectly orderable, weakly chordal, perfectly contractile, and chi-bound graphs. For each of these classes, when appropriate, we discuss the complexity of the recognition algorithm and algorithms for finding a minimum coloring, and a largest clique in the graph and its complement
Influence of the tie-break rule on the end-vertex problem
International audienceEnd-vertices of a given graph search may have some nice properties, as for example it is well known that the last vertex of Lexicographic Breadth First Search (LBFS) in a chordal graph is simplicial, see Rose, Tarjan and Lueker 1976. Therefore it is interesting to consider if these vertices can be recognized in polynomial time or not, as first studied in Corneil, Köhler and Lanlignel 2010. A graph search is a mechanism for systematically visiting the vertices of a graph. At each step of a graph search, the key point is the choice of the next vertex to be explored. Graph searches only differ by this selection mechanism during which a tie-break rule is used. In this paper we study how the choice of the tie-break in case of equality during the search, for a given graph search including the classic ones such as BFS and DFS, can determine the complexity of the end-vertex problem. In particular we prove a counterintuitive NP-completeness result for Breadth First Search solving a problem raised in Corneil, Köhler and Lanlignel 2010
Modelos para sequenciação de padrões em problemas de corte de stock
Tese de doutoramento em Engenharia Industrial e de SistemasIn this thesis, we address an optimization problem that appears in cutting stock
operations research called the minimization of the maximum number of open
stacks (MOSP) and we put forward a new integer programming formulation for
the MOSP.
By associating the duration of each stack with an interval of time, it is
possible to use the rich theory that exists in interval graphs in order to create
a model based on the completion of a graph with edges. The structure of this
type of graphs admits a linear ordering of the vertices that de nes an ordering
of the stacks, and consequently decides a sequence for the cutting patterns.
The polytope de ned by this formulation is full-dimensional and the main
inequalities in the model are proved to be facets. Additional inequalities are
derived based on the properties of chordal graphs and comparability graphs.
The maximum number of open stacks is related with the chromatic number of
the solution graph; thus the formulation is strengthened by adding the representatives
formulation for the vertex coloring problem.
The model is applied to the minimization of open stacks, and also to the
minimum interval graph completion problem and other pattern sequencing problems
such as the minimization of the order spread (MORP) and the minimization
of the number of tool switches (MTSP). Computational tests of the model are
presented.Nesta tese e abordado um problema de optimização que surge em operações de corte de stock chamado minimização do número máximo de pilhas abertas (MOSP) e e proposta uma nova formulação de programação inteira. Associando a duração de cada pilha a um intervalo de tempo, e possÃvel usar a teoria rica que existe em grafos de intervalos para criar um modelo baseado no completamento de um grafo por arcos. A estrutura deste tipo de grafos admite uma ordenação linear dos vértices que define uma ordenação linear das pilhas e, por sua vez, determina a sequência dos padrões de corte. O politopo definido por esta formulação tem dimensão completa e prova-se que as principais desigualdades do modelo são facetas. São derivadas desigualdades adicionais baseadas nas propriedades de grafos cordais e de grafos de comparabilidades. O número máximo de pilhas abertas está relacionado com o número cromático do grafo solução, pelo que o modelo e reforçado com a formulação por representativos para o problema de coloração de vértices. O modelo e aplicado a minimização de pilhas abertas, e também ao problema de completamento mÃnimo de um grafo de intervalos e a outros problemas de sequenciação de padrões, tais como a minimização da dispersão de encomendas (MORP) e a minimização do número de trocas de ferramentas (MTSP). São apresentados testes computacionais do modelo.Fundação para a Ciência e a Tecnologia (FCT),
programa de financiamento QREN-POPH-Tipologia 4.1-Formação Avançada comparticipado pelo Fundo Social Europeu e por fundos do MCTES
(Bolsa individual com a refer^encia SFRH/BD/32151/2006) entre 2006 e 2009, e
pela Escola Superior de Estudos Industriais e de Gest~ao do Instituto Polit ecnico
do Porto (Bolsa PROTEC com a refer^encia SFRH/BD/49914/2009) entre 2009
e 2010
Exploiting graph structures for computational efficiency
Coping with NP-hard graph problems by doing better than simply brute force is a field of significant practical importance, and which have also sparked wide theoretical interest. One route to cope with such hard graph problems is to exploit structures which can possibly be found in the input data or in the witness for a solution. In the framework of parameterized complexity, we attempt to quantify such structures by defining numbers which describe "how structured" the graph is. We then do a fine-grained classification of its computational complexity, where not only the input size, but also the structural measure in question come in to play. There is a number of structural measures called width parameters, which includes treewidth, clique-width, and mim-width. These width parameters can be compared by how many classes of graphs that have bounded width. In general there is a tradeoff; if more graph classes have bounded width, then fewer problems can be efficiently solved with the aid of a small width; and if a width is bounded for only a few graph classes, then it is easier to design algorithms which exploit the structure described by the width parameter. For each of the mentioned width parameters, there are known meta-theorems describing algorithmic results for a wide array of graph problems. Hence, showing that decompositions with bounded width can be found for a certain graph class yields algorithmic results for the given class. In the current thesis, we show that several graph classes have bounded width measures, which thus gives algorithmic consequences. Algorithms which are FPT or XP parameterized by width parameters are exploiting structure of the input graph. However, it is also possible to exploit structures that are required of a witness to the solution. We use this perspective to give a handful of polynomial-time algorithms for NP-hard problems whenever the witness belongs to certain graph classes. It is also possible to combine structures of the input graph with structures of the solution witnesses in order to obtain parameterized algorithms, when each structure individually is provably insufficient to provide so under standard complexity assumptions. We give an example of this in the final chapter of the thesis
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