2,013 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
Algorithmic Aspects of a General Modular Decomposition Theory
A new general decomposition theory inspired from modular graph decomposition
is presented. This helps unifying modular decomposition on different
structures, including (but not restricted to) graphs. Moreover, even in the
case of graphs, the terminology ``module'' not only captures the classical
graph modules but also allows to handle 2-connected components, star-cutsets,
and other vertex subsets. The main result is that most of the nice algorithmic
tools developed for modular decomposition of graphs still apply efficiently on
our generalisation of modules. Besides, when an essential axiom is satisfied,
almost all the important properties can be retrieved. For this case, an
algorithm given by Ehrenfeucht, Gabow, McConnell and Sullivan 1994 is
generalised and yields a very efficient solution to the associated
decomposition problem
Longest Common Pattern between two Permutations
In this paper, we give a polynomial (O(n^8)) algorithm for finding a longest
common pattern between two permutations of size n given that one is separable.
We also give an algorithm for general permutations whose complexity depends on
the length of the longest simple permutation involved in one of our
permutations
Some results on triangle partitions
We show that there exist efficient algorithms for the triangle packing
problem in colored permutation graphs, complete multipartite graphs,
distance-hereditary graphs, k-modular permutation graphs and complements of
k-partite graphs (when k is fixed). We show that there is an efficient
algorithm for C_4-packing on bipartite permutation graphs and we show that
C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite
graphs that have a triangle partition
Computing commons interval of K permutations, with applications to modular decomposition of graphs
International audienceWe introduce a new way to compute common intervals of K permutations based on a very simple and general notion of generators of common intervals. This formalism leads to simple and efficient algorithms to compute the set of all common intervals of K permutations, that can contain a quadratic number of intervals, as well as a linear space basis of this set of common intervals. Finally, we show how our results on permutations can be used for computing the modular decomposition of graphs in linear time
A general method for common intervals
Given an elementary chain of vertex set V, seen as a labelling of V by the
set {1, ...,n=|V|}, and another discrete structure over , say a graph G, the
problem of common intervals is to compute the induced subgraphs G[I], such that
is an interval of [1, n] and G[I] satisfies some property Pi (as for
example Pi= "being connected"). This kind of problems comes from comparative
genomic in bioinformatics, mainly when the graph is a chain or a tree
(Heber and Stoye 2001, Heber and Savage 2005, Bergeron et al 2008).
When the family of intervals is closed under intersection, we present here
the combination of two approaches, namely the idea of potential beginning
developed in Uno, Yagiura 2000 and Bui-Xuan et al 2005 and the notion of
generator as defined in Bergeron et al 2008. This yields a very simple generic
algorithm to compute all common intervals, which gives optimal algorithms in
various applications. For example in the case where is a tree, our
framework yields the first linear time algorithms for the two properties:
"being connected" and "being a path". In the case where is a chain, the
problem is known as: common intervals of two permutations (Uno and Yagiura
2000), our algorithm provides not only the set of all common intervals but also
with some easy modifications a tree structure that represents this set
Automorphism Groups of Geometrically Represented Graphs
We describe a technique to determine the automorphism group of a
geometrically represented graph, by understanding the structure of the induced
action on all geometric representations. Using this, we characterize
automorphism groups of interval, permutation and circle graphs. We combine
techniques from group theory (products, homomorphisms, actions) with data
structures from computer science (PQ-trees, split trees, modular trees) that
encode all geometric representations.
We prove that interval graphs have the same automorphism groups as trees, and
for a given interval graph, we construct a tree with the same automorphism
group which answers a question of Hanlon [Trans. Amer. Math. Soc 272(2), 1982].
For permutation and circle graphs, we give an inductive characterization by
semidirect and wreath products. We also prove that every abstract group can be
realized by the automorphism group of a comparability graph/poset of the
dimension at most four
The Longest Common Pattern Problem for two Permutations
International audienceIn this paper, we give a polynomial (O(n^8)) algorithm for finding a longest common pattern between two permutations of size n given that one is separable. We also give an algorithm for general permutations whose complexity depends on the length of the longest simple permutation involved in one of our permutations
Longest Common Separable Pattern between Permutations
In this article, we study the problem of finding the longest common separable
pattern between several permutations. We give a polynomial-time algorithm when
the number of input permutations is fixed and show that the problem is NP-hard
for an arbitrary number of input permutations even if these permutations are
separable. On the other hand, we show that the NP-hard problem of finding the
longest common pattern between two permutations cannot be approximated better
than within a ratio of (where is the size of an optimal
solution) when taking common patterns belonging to pattern-avoiding classes of
permutations.Comment: 15 page
Average-case analysis of perfect sorting by reversals (Journal Version)
Perfect sorting by reversals, a problem originating in computational
genomics, is the process of sorting a signed permutation to either the identity
or to the reversed identity permutation, by a sequence of reversals that do not
break any common interval. B\'erard et al. (2007) make use of strong interval
trees to describe an algorithm for sorting signed permutations by reversals.
Combinatorial properties of this family of trees are essential to the algorithm
analysis. Here, we use the expected value of certain tree parameters to prove
that the average run-time of the algorithm is at worst, polynomial, and
additionally, for sufficiently long permutations, the sorting algorithm runs in
polynomial time with probability one. Furthermore, our analysis of the subclass
of commuting scenarios yields precise results on the average length of a
reversal, and the average number of reversals.Comment: A preliminary version of this work appeared in the proceedings of
Combinatorial Pattern Matching (CPM) 2009. See arXiv:0901.2847; Discrete
Mathematics, Algorithms and Applications, vol. 3(3), 201
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