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 $O\left(n^2\right)$ decomposition algorithm and we take advantage of the involution modular decomposition tree to state several algorithmic results. Cographs are the graphs that are totally decomposable w.r.t modular decomposition. In a similar way, we introduce the class of switch cographs, the class of graphs that are totally decomposable w.r.t involution modular decomposition. This class generalizes the class of cographs and is exactly the class of (Bull, Gem, Co-Gem, $C_5$)-free graphs. We use our new decomposition tool to design three practical algorithms for the maximum cut, vertex cover and vertex separator problems. The complexity of these problems was still unknown for this class of graphs. This paper also improves the complexity of the maximum clique, the maximum independant set, the chromatic number and the maximum clique cover problems by giving efficient algorithms, thanks to the decomposition tree. Eventually, we show that this class of graphs has Clique-Width at most 4 and that a Clique-Width expression can be computed in linear time.

LDFS-Based Certifying Algorithm for the Minimum Path Cover Problem on Cocomparability Graphs

Participants : Derek Corneil, Dalton Barnaby, Michel Habib.

For graph $G(V,E)$, a minimum path cover (MPC) is a minimum cardinality set of vertex disjoint paths that cover $V$ (i.e., every vertex of $G$ is in exactly one path in the cover). This problem is a natural generalization of the Hamiltonian path problem. Cocomparability graphs (the complements of graphs that have an acyclic transitive orientation of their edge sets) are a well studied subfamily of perfect graphs that includes many popular families of graphs such as interval, permutation, and cographs. Furthermore, for every cocomparability graph $G$ and acyclic transitive orientation of the edges of $\overline{G}$ there is a corresponding poset $P_G$; it is easy to see that an MPC of $G$ is a linear extension of $P_G$ that minimizes the bump number of $P_G$. Although there are directly graph-theoretical MPC algorithms (i.e., algorithms that do not rely on poset formulations) for various subfamilies of cocomparability graphs, notably interval graphs, until now all MPC algorithms for cocomparability graphs themselves have been based on the bump number algorithms for posets. In this paper [5] , we present the first directly graph-theoretical MPC algorithm for cocomparability graphs; this algorithm is based on two consecutive graph searches followed by a certifying algorithm. Surprisingly, except for a lexicographic depth first search (LDFS) preprocessing step, this algorithm is identical to the corresponding algorithm for interval graphs. The running time of the algorithm is $O\left(\min (n^2,n+\mathrm{mloglogn})\right)$, with the nonlinearity coming from LDFS.

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 $K$ genomes represented as $K$ permutations. A $b-$nested common (resp. conserved) interval $I$ of size $\left|I\right|$ is either an interval of size 1 or a common (resp. conserved) interval that contains another b-nested common (resp. conserved) interval of size at least $\left|I\right|-b$. When $b=1$, this corresponds to the classical notion of nested interval. We exhibit two simple algorithms to output all b-nested common or conserved intervals between $K$ permutations in $O(Kn+\text{nocc})$ time, where nocc is the total number of such intervals. We also explain how to count all b-nested intervals in O(Kn) time. New properties of the family of conserved intervals are proposed to do so.

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 $(\Delta +1)$-coloring algorithms by Barenboim and Elkin, by Kuhn, and by Panconesi and Srinivasan, as well as the $o\left({\Delta }^2\right)$-coloring algorithm by Linial. Unfortunately, most known local algorithms (including, in particular, the aforementioned algorithms) are non-uniform, that is, they assume that all nodes know good estimations of one or more global parameters of the network, e.g., the maximum degree $\Delta$ or the number of nodes $n$. This paper [11] , provides a rather general method for transforming a non-uniform local algorithm into a uniform one. Furthermore, the resulting algorithm enjoys the same asymptotic running time as the original non-uniform algorithm. Our method applies to a wide family of both deterministic and randomized algorithms. Specifically, it applies to almost all of the state of the art non-uniform algorithms regarding MIS and Maximal Matching, as well as to many results concerning the coloring problem. (In particular, it applies to all aforementioned algorithms.) To obtain our transformations we introduce a new distributed tool called pruning algorithms, which we believe may be of independent interest.

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.