96 research outputs found
A survey on algorithmic aspects of modular decomposition
The modular decomposition is a technique that applies but is not restricted
to graphs. The notion of module naturally appears in the proofs of many graph
theoretical theorems. Computing the modular decomposition tree is an important
preprocessing step to solve a large number of combinatorial optimization
problems. Since the first polynomial time algorithm in the early 70's, the
algorithmic of the modular decomposition has known an important development.
This paper survey the ideas and techniques that arose from this line of
research
Representation of graphs by OBDDs
AbstractRecently, it has been shown in a series of works that the representation of graphs by Ordered Binary Decision Diagrams (OBDDs) often leads to good algorithmic behavior. However, the question for which graph classes an OBDD representation is advantageous, has not been investigated, yet. In this paper, the space requirements for the OBDD representation of certain graph classes, specifically cographs, several types of graphs with few P4s, unit interval graphs, interval graphs and bipartite graphs are investigated. Upper and lower bounds are proven for all these graph classes and it is shown that in most (but not all) cases a representation of the graphs by OBDDs is advantageous with respect to space requirements
Fully polynomial FPT algorithms for some classes of bounded clique-width graphs
Parameterized complexity theory has enabled a refined classification of the
difficulty of NP-hard optimization problems on graphs with respect to key
structural properties, and so to a better understanding of their true
difficulties. More recently, hardness results for problems in P were achieved
using reasonable complexity theoretic assumptions such as: Strong Exponential
Time Hypothesis (SETH), 3SUM and All-Pairs Shortest-Paths (APSP). According to
these assumptions, many graph theoretic problems do not admit truly
subquadratic algorithms, nor even truly subcubic algorithms (Williams and
Williams, FOCS 2010 and Abboud, Grandoni, Williams, SODA 2015). A central
technique used to tackle the difficulty of the above mentioned problems is
fixed-parameter algorithms for polynomial-time problems with polynomial
dependency in the fixed parameter (P-FPT). This technique was introduced by
Abboud, Williams and Wang in SODA 2016 and continued by Husfeldt (IPEC 2016)
and Fomin et al. (SODA 2017), using the treewidth as a parameter. Applying this
technique to clique-width, another important graph parameter, remained to be
done. In this paper we study several graph theoretic problems for which
hardness results exist such as cycle problems (triangle detection, triangle
counting, girth, diameter), distance problems (diameter, eccentricities, Gromov
hyperbolicity, betweenness centrality) and maximum matching. We provide
hardness results and fully polynomial FPT algorithms, using clique-width and
some of its upper-bounds as parameters (split-width, modular-width and
-sparseness). We believe that our most important result is an -time algorithm for computing a maximum matching where
is either the modular-width or the -sparseness. The latter generalizes
many algorithms that have been introduced so far for specific subclasses such
as cographs, -lite graphs, -extendible graphs and -tidy
graphs. Our algorithms are based on preprocessing methods using modular
decomposition, split decomposition and primeval decomposition. Thus they can
also be generalized to some graph classes with unbounded clique-width
Computing Well-Covered Vector Spaces of Graphs using Modular Decomposition
A graph is well-covered if all its maximal independent sets have the same
cardinality. This well studied concept was introduced by Plummer in 1970 and
naturally generalizes to the weighted case. Given a graph , a real-valued
vertex weight function is said to be a well-covered weighting of if all
its maximal independent sets are of the same weight. The set of all
well-covered weightings of a graph forms a vector space over the field of
real numbers, called the well-covered vector space of . Since the problem of
recognizing well-covered graphs is --complete, the
problem of computing the well-covered vector space of a given graph is
--hard. Levit and Tankus showed in 2015 that the
problem admits a polynomial-time algorithm in the class of claw-free graph. In
this paper, we give two general reductions for the problem, one based on
anti-neighborhoods and one based on modular decomposition, combined with
Gaussian elimination. Building on these results, we develop a polynomial-time
algorithm for computing the well-covered vector space of a given fork-free
graph, generalizing the result of Levit and Tankus. Our approach implies that
well-covered fork-free graphs can be recognized in polynomial time and also
generalizes some known results on cographs.Comment: 25 page
Distributed Certification for Classes of Dense Graphs
A proof-labeling scheme (PLS) for a boolean predicate on labeled graphs
is a mechanism used for certifying the legality with respect to of global
network states in a distributed manner. In a PLS, a certificate is assigned to
each processing node of the network, and the nodes are in charge of checking
that the collection of certificates forms a global proof that the system is in
a correct state, by exchanging the certificates once, between neighbors only.
The main measure of complexity is the size of the certificates. Many PLSs have
been designed for certifying specific predicates, including cycle-freeness,
minimum-weight spanning tree, planarity, etc.
In 2021, a breakthrough has been obtained, as a meta-theorem stating that a
large set of properties have compact PLSs in a large class of networks. Namely,
for every property on labeled graphs, there exists a PLS
for with -bit certificates for all graphs of bounded
tree-depth. This result has been extended to the larger class of graphs with
bounded {tree-width}, using certificates on bits.
We extend this result even further, to the larger class of graphs with
bounded clique-width, which, as opposed to the other two aforementioned
classes, includes dense graphs. We show that, for every
property on labeled graphs, there exists a PLS for with bit certificates for all graphs of bounded clique-width
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