Section: New Results
Understanding graph representations
Connected graph searching
Computing H-Joins with Application to 2-Modular Decomposition
Participants : Michel Habib, Antoine Mamcarz, Fabien de Montgolfier.
We present in [10] , a general framework to design algorithms that compute H-join. For a given bipartite graph H, we say that a graph G admits a H-join decomposition or simply a H-join, if the vertices of G can be partitioned in |H| parts connected as in H. This graph H is a kind of pattern, that we want to discover in G. This framework allows us to present fastest known algorithms for the computation of P 4-join (aka N-join), P 5-join (aka W-join), C 6-join (aka 6-join). We also generalize this method to find a homogeneous pair (also known as 2-module), a pair M 1,M 2 such that for every vertex x∉(M 1∪M 2) and i∈1,2, x is either adjacent to all vertices in M i or to none of them. First used in the context of perfect graphs (Chvátal and Sbihi in Graphs Comb. 3:127-139, 1987), it is a generalization of splits (a.k.a. 1-joins) and of modules. The algorithmics to compute them appears quite involved. In this paper, we describe an O(mn 2)-time algorithm computing all maximal homogeneous pairs of a graph, which not only improves a previous bound of O(mn 3) for finding only one pair (Everett et al. in Discrete Appl. Math. 72:209-218, 1997), but also uses a nice structural property of homogenous pairs, allowing to compute a canonical decomposition tree for sesquiprime graphs (i.e., graphs G having no module and such that for every vertex v∈G, G−v also has no module).
Algorithmic Aspects of Switch Cographs
Participants : Vincent Cohen-Addad, Michel Habib, Fabien de Montgolfier.
The paper [27] , introduces the notion of
involution
module, the first generalization of the modular decomposition of
2-structure
which has a unique linear-sized decomposition tree.
We derive an
LDFS-Based Certifying Algorithm for the Minimum Path Cover Problem on Cocomparability Graphs
Participants : Derek Corneil, Dalton Barnaby, Michel Habib.
For graph
Easy identification of generalized common and conserved nested intervals
Participants : Fabien de Montgolfier, Mathieu Raffinot, Irena Rusu.
In the paper [28] , we explain how to easily compute gene clusters,
formalized by classical or generalized nested common or conserved
intervals, between a set of
On computing the diameter of real-world undirected graphs
Participants : Pierluigi Crescenzi, Roberto Grossi, Michel Habib, Leonardo Lanzi, Andrea Marino.
We propose in [2] ,a new algorithm for the classical problem of computing the diameter of undirected unweighted graphs, namely, the maximum distance among all the pairs of nodes, where the distance of a pair of nodes is the number of edges contained in the shortest path connecting these two nodes. Although its worst-case complexity is O(nm) time, where n is the number of nodes and m is the number of edges of the graph, we experimentally show that our algorithm works in O(m) time in practice, requiring few breadth-first searches to complete its task on almost 200 real-world graphs.
Toward more localized local algorithms: removing assumptions concerning global knowledge
Participants : Amos Korman, Jean-Sébastien Sereni, Laurent Viennot.
Numerous sophisticated local algorithm were suggested in the
literature for various fundamental problems. Notable examples are the
MIS and
Self-organizing Flows in Social Networks
Participants : Nidhi Hegde, Laurent Massoulié, Laurent Viennot.
Social networks offer users new means of accessing information, essentially relying on ”social filtering”, i.e. propagation and filtering of information by social contacts. The sheer amount of data flowing in these networks, combined with the limited budget of attention of each user, makes it difficult to ensure that social filtering brings relevant content to the interested users. Our motivation in this paper [24] , is to measure to what extent self-organization of the social network results in efficient social filtering. To this end we introduce flow games, a simple abstraction that models network formation under selfish user dynamics, featuring user-specific interests and budget of attention. In the context of homogeneous user interests, we show that selfish dynamics converge to a stable network structure (namely a pure Nash equilibrium) with close-to-optimal information dissemination. We show in contrast, for the more realistic case of heterogeneous interests, that convergence, if it occurs, may lead to information dissemination that can be arbitrarily inefficient, as captured by an unbounded ”price of anarchy”. Nevertheless the situation differs when users' interests exhibit a particular structure, captured by a metric space with low doubling dimension. In that case, natural autonomous dynamics converge to a stable configuration. Moreover, users obtain all the information of interest to them in the corresponding dissemination, provided their budget of attention is logarithmic in the size of their interest set.