Generalizations and Strengthenings of Ryser's Conjecture

Ryser's conjecture says that for every $r$-partite hypergraph $H$ with matching number $\nu(H)$, the vertex cover number is at most $(r-1)\nu(H)$. This far-reaching generalization of König's theorem is only known to be true for $r\leq 3$, or when $\nu(H)=1$ and $r\leq 5$. An equivalent formulation of Ryser's conjecture is that in every $r$-edge coloring of a graph $G$ with independence number $\alpha(G)$, there exists at most $(r-1)\alpha(G)$ monochromatic connected subgraphs which cover the vertex set of $G$. We make the case that this latter formulation of Ryser's conjecture naturally leads to a variety of stronger conjectures and generalizations to hypergraphs and multipartite graphs. Regarding these generalizations and strengthenings, we survey the known results, improving upon some, and we introduce a collection of new problems and results.

CALDERA: Finding all significant de Bruijn subgraphs for bacterial GWAS

Genome wide association studies (GWAS), aiming to find genetic variants associated with a trait, have widely been used on bacteria to identify genetic determinants of drug resistance or hypervirulence. Recent bacterial GWAS methods usually rely on k-mers, whose presence in a genome can denote variants ranging from single nucleotide polymorphisms to mobile genetic elements. Since many bacterial species include genes that are not shared among all strains, this approach avoids the reliance on a common reference genome. However, the same gene can exist in slightly different versions across different strains, leading to diluted effects when trying to detect its association to a phenotype through k-mer based GWAS. Here we propose to overcome this by testing covariates built from closed connected subgraphs of the De Bruijn graph defined over genomic k-mers. These covariates are able to capture polymorphic genes as a single entity, improving k-mer based GWAS in terms of power and interpretability. As the number of subgraphs is exponential in the number of nodes in the DBG, a method naively testing all possible subgraphs would result in very low statistical power due to multiple testing corrections, and the mere exploration of these subgraphs would quickly become computationally intractable. The concept of testable hypothesis has successfully been used to address both problems in similar contexts. We leverage this concept to test all closed connected subgraphs by proposing a novel enumeration scheme for these objects which fully exploits the pruning opportunity offered by testability, resulting in drastic improvements in computational efficiency. We illustrate this on both real and simulated datasets and also demonstrate how considering subgraphs leads to a more powerful and interpretable method. Our method integrates with existing visual tools to facilitate interpretation. We also provide an implementation of our method, as well as code to reproduce all results at https://github.com/HectorRDB/Caldera_Recomb.

Using dual-network-analyser for communities detecting in dual networks

Background Representations of the relationships among data using networks are widely used in several research fields such as computational biology, medical informatics and social network mining. Recently, complex networks have been introduced to better capture the insights of the modelled scenarios. Among others, dual networks (DNs) consist of mapping information as pairs of networks containing the same set of nodes but with different edges: one, called physical network, has unweighted edges, while the other, called conceptual network, has weighted edges. Results We focus on DNs and we propose a tool to find common subgraphs (aka communities) in DNs with particular properties. The tool, called Dual-Network-Analyser, is based on the identification of communities that induce optimal modular subgraphs in the conceptual network and connected subgraphs in the physical one. It includes the Louvain algorithm applied to the considered case. The Dual-Network-Analyser can be used to study DNs, to find common modular communities. We report results on using the tool to identify communities on synthetic DNs as well as real cases in social networks and biological data. Conclusion The proposed method has been tested by using synthetic and biological networks. Results demonstrate that it is well able to detect meaningful information from DNs.

Stop Bickering! Reconciling Signaling Pathway Databases with Network Topologies

A major goal of molecular systems biology is to understand the coordinated function of genes or proteins in response to cellular signals and to understand these dynamics in the context of disease. Signaling pathway databases such as KEGG, NetPath, NCI-PID, and Panther describe the molecular interactions involved in different cellular responses. While the same pathway may be present in different databases, prior work has shown that the particular proteins and interactions differ across database annotations. However, to our knowledge no one has attempted to quantify their structural differences. It is important to characterize artifacts or other biases within pathway databases, which can provide a more informed interpretation for downstream analyses. In this work, we consider signaling pathways as graphs and we use topological measures to study their structure. We find that topological characterization using graphlets (small, connected subgraphs) distinguishes signaling pathways from appropriate null models of interaction networks. Next, we quantify topological similarity across pathway databases. Our analysis reveals that the pathways harbor database-specific characteristics implying that even though these databases describe the same pathways, they tend to be systematically different from one another. We show that pathway-specific topology can be uncovered after accounting for database-specific structure. This work present the first step towards elucidating common pathway structure beyond their specific database annotations.

The Balanced Connected k-Partition Problem: Polyhedra and Algorithms

The balanced connected k-partition (BCPk) problem consists in partitioning a connected graph into connected subgraphs with similar weights. This problem arises in multiple practical applications, such as police patrolling, image processing, data base and operating systems. In this work, we address the BCPk using mathematical programming. We propose a compact formulation based on flows and a formulation based on separators. We introduce classes of valid inequalities and design polynomial-time separation routines. Moreover, to the best of our knowledge, we present the first polyhedral study for BCPk in the literature. Finally, we report on computational experiments showing that the proposed algorithms significantly outperform the state of the art for BCPk.

Butterfly-core community search over labeled graphs

Community search aims at finding densely connected subgraphs for query vertices in a graph. While this task has been studied widely in the literature, most of the existing works only focus on finding homogeneous communities rather than heterogeneous communities with different labels. In this paper, we motivate a new problem of cross-group community search, namely Butterfly-Core Community (BCC), over a labeled graph, where each vertex has a label indicating its properties and an edge between two vertices indicates their cross relationship. Specifically, for two query vertices with different labels, we aim to find a densely connected cross community that contains two query vertices and consists of butterfly networks, where each wing of the butterflies is induced by a k-core search based on one query vertex and two wings are connected by these butterflies. We first develop a heuristic algorithm achieving 2-approximation to the optimal solution. Furthermore, we design fast techniques of query distance computations, leader pair identifications, and index-based BCC local explorations. Extensive experiments on seven real datasets and four useful case studies validate the effectiveness and efficiency of our BCC and its multi-labeled extension models.