On the complexity of finding a potential community

An independent 2-clique of a graph is a subset of vertices that is an independent set and such that any two vertices inside have a common neighbor outside. In this paper, we study the complexity of find-ing an independent 2-clique of maximum size in several graph classes and we compare its complexity with the complexity of maximum independent set. We prove that this problem is NP-hard on apex graphs, APX-hard on line graphs, not n1 /2−-approximable on bipartite graphs and not-approximable on split graphs, while it is polynomial-time solvable on graphs of bounded degree and their complements, graphs of bounded treewidth, planar graphs, (C3, C6)-free graphs, threshold graphs, interval graphs and cographs. © Springer International Publishing AG 2017

Détection de communautés : complexité computationnelle et approximation

This thesis deals with community detection in the context of social networks. A social network can be modeled by a graph in which vertices represent members, and edges represent relationships. In particular, I study four different definitions of a community. First, a community structure can be defined as a partition of the vertices such that each vertex has a greater proportion of neighbors in its part than in any other part. This definition can be adapted in order to study only one community. Then, a community can be viewed as a subgraph in which every two vertices are at distance 2 in this subgraph. Finally, in the context of online meetup services, I investigate a definition for potential communities in which members do not know each other but are related by their common neighbors. In regard to these proposed definitions, I study computational complexity and approximation within problems that either relate to the existence of such communities or to finding them in graphs.Cette thèse étudie la détection de communautés dans le contexte des réseaux sociaux. Un réseau social peut être modélisé par un graphe dans lequel les sommets représentent les membres et les arêtes représentent les relations entre les membres. En particulier, j'étudie quatre différentes définitions de communauté. D'abord, une structure en communautés peut être définie par une partition des sommets telle que tout sommet a une plus grande proportion de voisins dans sa partie que dans toute autre partie. Cette définition peut être adaptée pour l'étude d'une seule communauté. Ensuite, une communauté peut être vue comme un sous graphe tel que tout couple de sommets sont à distance 2 dans ce sous graphe. Enfin, dans le contexte des sites de rencontre, je propose d'étudier une définition de communauté potentielle dans le sens où les membres de la communauté ne se connaissent pas, mais sont liés par des connaissances communes. Pour ces trois définitions, j'étudie la complexité computationnelle et l'approximation de problèmes liés à l'existence ou la recherche de telles communautés dans les graphes

Finding a potential community in networks

An independent 2-clique of a graph is a subset of vertices that is an independent set and such that any two vertices inside have a common neighbor outside. In this paper, we study the complexity of finding an independent 2-clique of maximum size in several graph classes and we compare it with the complexity of maximum independent set

Proportionally dense subgraph of maximum size

We define a proportionally dense subgraph (PDS) as an induced subgraph of a graph with the property that each vertex in the PDS is adjacent to proportionally as many vertices in the subgraph as in the graph. We prove that the problem of finding a PDS of maximum size is APX-hard on split graphs, and NP-hard on bipartite graphs. We also show that deciding if a PDS is inclusion-wise maximal is co-NP-complete on bipartite graphs. Nevertheless, we present a simple polynomial-time $(2-\frac{2}{\Delta+1})$-approximation algorithm for the problem, where $\Delta$ is the maximum degree of the graph. Finally, we show that all Hamiltonian cubic graphs with $n$ vertices (except two) have a PDS of size $\lfloor \frac{2n+1}{3} \rfloor$, which we prove to be an upper bound on the size of a PDS in cubic graphs

Biallelic Mutations in LIPT2 Cause a Mitochondrial Lipoylation Defect Associated with Severe Neonatal Encephalopathy

Lipoate serves as a cofactor for the glycine cleavage system (GCS) and four 2-oxoacid dehydrogenases functioning in energy metabolism (α-oxoglutarate dehydrogenase [α-KGDHc] and pyruvate dehydrogenase [PDHc]), or amino acid metabolism (branched-chain oxoacid dehydrogenase, 2-oxoadipate dehydrogenase). Mitochondrial lipoate synthesis involves three enzymatic steps catalyzed sequentially by lipoyl(octanoyl) transferase 2 (LIPT2), lipoic acid synthetase (LIAS), and lipoyltransferase 1 (LIPT1). Mutations in LIAS have been associated with nonketotic hyperglycinemia-like early-onset convulsions and encephalopathy combined with a defect in mitochondrial energy metabolism. LIPT1 deficiency spares GCS deficiency and has been associated with a biochemical signature of combined 2-oxoacid dehydrogenase deficiency leading to early death or Leigh-like encephalopathy. We report on the identification of biallelic LIPT2 mutations in three affected individuals from two families with severe neonatal encephalopathy. Brain MRI showed major cortical atrophy with white matter abnormalities and cysts. Plasma glycine was mildly increased. Affected individuals' fibroblasts showed reduced oxygen consumption rates, PDHc, α-KGDHc activities, leucine catabolic flux, and decreased protein lipoylation. A normalization of lipoylation was observed after expression of wild-type LIPT2, arguing for LIPT2 requirement in intramitochondrial lipoate synthesis. Lipoic acid supplementation did not improve clinical condition nor activities of PDHc, α-KGDHc, or leucine metabolism in fibroblasts and was ineffective in yeast deleted for the orthologous LIP2