13 research outputs found
Clique coloring -EPG graphs
We consider the problem of clique coloring, that is, coloring the vertices of
a given graph such that no (maximal) clique of size at least two is
monocolored. It is known that interval graphs are -clique colorable. In this
paper we prove that -EPG graphs (edge intersection graphs of paths on a
grid, where each path has at most one bend) are -clique colorable. Moreover,
given a -EPG representation of a graph, we provide a linear time algorithm
that constructs a -clique coloring of it.Comment: 9 Page
B1-EPG graphs are 4-clique colorable
We consider the problem of clique coloring, that is, coloring the vertices of a given graph such that no (maximal) clique of size at least two is monocolored. It is known that interval graphs are 2-clique colorable. In this work we prove that B1-EPG graphs (edge intersection graphs of paths on a grid, where each path has at most one bend) are 4-clique colorable. Moreover, given a B1-EPG representation of a graph, we provide a linear time algorithm that constructs a 4-clique coloring of it.Facultad de Ciencias Exacta
B1-EPG graphs are 4-clique colorable
We consider the problem of clique coloring, that is, coloring the vertices of a given graph such that no (maximal) clique of size at least two is monocolored. It is known that interval graphs are 2-clique colorable. In this work we prove that B1-EPG graphs (edge intersection graphs of paths on a grid, where each path has at most one bend) are 4-clique colorable. Moreover, given a B1-EPG representation of a graph, we provide a linear time algorithm that constructs a 4-clique coloring of it.Facultad de Ciencias Exacta
Clique coloring B1-EPG graphs
We consider the problem of clique coloring, that is, coloring the vertices of a given graph such that no (maximal) clique of size at least two is monocolored. It is known that interval graphs are 2-clique colorable. In this paper we prove that B1-EPG graphs (edge intersection graphs of paths on a grid, where each path has at most one bend) are 4-clique colorable.
Moreover, given a B1-EPG representation of a graph, we provide a linear time algorithm that constructs a 4-clique coloring of it.Facultad de Ciencias Exacta
B1-EPG graphs are 4-clique colorable
We consider the problem of clique coloring, that is, coloring the vertices of a given graph such that no (maximal) clique of size at least two is monocolored. It is known that interval graphs are 2-clique colorable. In this work we prove that B1-EPG graphs (edge intersection graphs of paths on a grid, where each path has at most one bend) are 4-clique colorable. Moreover, given a B1-EPG representation of a graph, we provide a linear time algorithm that constructs a 4-clique coloring of it.Facultad de Ciencias Exacta
Sobre la clique coloración de los grafos [4,2,2]
Un grafo de intersección por aristas de una familia de caminos en un árbol huesped es llamado grafo EPT. Cuando el grado máximo del árbol huesped es 4, decimos que el grafo es [4, 2, 2]. En este trabajo, consideramos el problema de clique coloración en grafos [4,2, 2]. Probamos que esta clase de grafos es 3-clique coloreable y damos ejemplos de grafos en esta clase que no son 2-clique coloreables. Además, estudiamos subclases de grafos en [4, 2,2] que tienen número clique cromático menor o igual a 2.Centro de Investigación de Matemátic
On the helly property of some intersection graphs
An EPG graph G is an edge-intersection graph of paths on a grid. In this
doctoral thesis we will mainly explore the EPG graphs, in particular B1-EPG graphs.
However, other classes of intersection graphs will be studied such as VPG, EPT and
VPT graph classes, in addition to the parameters Helly number and strong Helly
number to EPG and VPG graphs. We will present the proof of NP-completeness
to Helly-B1-EPG graph recognition problem. We investigate the parameters Helly
number and the strong Helly number in both graph classes, EPG and VPG in order
to determine lower bounds and upper bounds for this parameters. We completely
solve the problem of determining the Helly and strong Helly numbers, for Bk-EPG,
and Bk-VPG graphs, for each value k.
Next, we present the result that every Chordal B1-EPG graph is simultaneously
in the VPT and EPT graph classes. In particular, we describe structures that occur
in B1-EPG graphs that do not support a Helly-B1-EPG representation and thus we
define some sets of subgraphs that delimit Helly subfamilies. In addition, features
of some non-trivial graph families that are properly contained in Helly-B1 EPG are
also presented.EPG é um grafo de aresta-interseção de caminhos sobre uma grade.
Nesta tese de doutorado exploraremos principalmente os grafos EPG, em particular
os grafos B1-EPG. Entretanto, outras classes de grafos de interseção serão estu dadas, como as classes de grafos VPG, EPT e VPT, além dos parâmetros número
de Helly e número de Helly forte nos grafos EPG e VPG. Apresentaremos uma
prova de NP-completude para o problema de reconhecimento de grafos B1-EPG Helly. Investigamos os parâmetros número de Helly e o número de Helly forte nessas
duas classes de grafos, EPG e VPG, a fim de determinar limites inferiores e superi ores para esses parâmetros. Resolvemos completamente o problema de determinar o
número de Helly e o número de Helly forte para os grafos Bk-EPG e Bk-VPG, para
cada valor k.
Em seguida, apresentamos o resultado de que todo grafo B1-EPG Chordal está
simultaneamente nas classes de grafos VPT e EPT. Em particular, descrevemos
estruturas que ocorrem em grafos B1-EPG que não suportam uma representação
B1-EPG-Helly e assim definimos alguns conjuntos de subgrafos que delimitam sub famÃlias Helly. Além disso, também são apresentadas caracterÃsticas de algumas
famÃlias de grafos não triviais que estão propriamente contidas em B1-EPG-Hell
Algorithms to Approximate Column-Sparse Packing Problems
Column-sparse packing problems arise in several contexts in both
deterministic and stochastic discrete optimization. We present two unifying
ideas, (non-uniform) attenuation and multiple-chance algorithms, to obtain
improved approximation algorithms for some well-known families of such
problems. As three main examples, we attain the integrality gap, up to
lower-order terms, for known LP relaxations for k-column sparse packing integer
programs (Bansal et al., Theory of Computing, 2012) and stochastic k-set
packing (Bansal et al., Algorithmica, 2012), and go "half the remaining
distance" to optimal for a major integrality-gap conjecture of Furedi, Kahn and
Seymour on hypergraph matching (Combinatorica, 1993).Comment: Extended abstract appeared in SODA 2018. Full version in ACM
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