Enumerating Maximal Induced Subgraphs

Given a graph G, the maximal induced subgraphs problem asks to enumerate all maximal induced subgraphs of G that belong to a certain hereditary graph class. While its optimization version, known as the minimum vertex deletion problem in literature, has been intensively studied, enumeration algorithms were only known for a few simple graph classes, e.g., independent sets, cliques, and forests, until very recently [Conte and Uno, STOC 2019]. There is also a connected variation of this problem, where one is concerned with only those induced subgraphs that are connected. We introduce two new approaches, which enable us to develop algorithms that solve both variations for a number of important graph classes. A general technique that has been proven very powerful in enumeration algorithms is to build a solution map, i.e., a multiple digraph on all the solutions of the problem, and the key of this approach is to make the solution map strongly connected, so that a simple traversal of the solution map solves the problem. First, we introduce retaliation-free paths to certify strong connectedness of the solution map we build. Second, generalizing the idea of Cohen, Kimelfeld, and Sagiv [JCSS 2008], we introduce an apparently very restricted version of the maximal (connected) induced subgraphs problem, and show that it is equivalent to the original problem in terms of solvability in incremental polynomial time. Moreover, we give reductions between the two variations, so that it suffices to solve one of the variations for each class we study. Our work also leads to direct and simpler proofs of several important known results

GPD: A Graph Pattern Diffusion Kernel for Accurate Graph Classification with Applications in Cheminformatics

Graph data mining is an active research area. Graphs are general modeling tools to organize information from heterogeneous sources and have been applied in many scientific, engineering, and business fields. With the fast accumulation of graph data, building highly accurate predictive models for graph data emerges as a new challenge that has not been fully explored in the data mining community. In this paper, we demonstrate a novel technique called graph pattern diffusion (GPD) kernel. Our idea is to leverage existing frequent pattern discovery methods and to explore the application of kernel classifier (e.g., support vector machine) in building highly accurate graph classification. In our method, we first identify all frequent patterns from a graph database. We then map subgraphs to graphs in the graph database and use a process we call “pattern diffusion” to label nodes in the graphs. Finally, we designed a graph alignment algorithm to compute the inner product of two graphs. We have tested our algorithm using a number of chemical structure data. The experimental results demonstrate that our method is significantly better than competing methods such as those kernel functions based on paths, cycles, and subgraphs

ZERO-KNOWLEDGE DE NOVO ALGORITHMS FOR ANALYZING SMALL MOLECULES USING MASS SPECTROMETRY

In the analysis of mass spectra, if a superset of the molecules thought to be in a sample is known a priori, then there are well established techniques for the identification of the molecules such as database search and spectral libraries. Linear molecules are chains of subunits. For example, a peptide is a linear molecule with an “alphabet” of 20 possible amino acid subunits. A peptide of length six will have 206 = 64, 000, 000 different possible outcomes. Small molecules, such as sugars and metabolites, are not constrained to linear structures and may branch. These molecules are encoded as undirected graphs rather than simply linear chains. An undirected graph with six subunits (each of which have 20 possible outcomes) will 6 have 206 · 2(6 choose 2) = 2, 097, 152, 000, 000 possible outcomes. The vast amount of complex graphs which small molecules can form can render databases and spectral libraries impossibly large to use or incomplete as many metabolites may still be unidentified. In the absence of a usable database or spectral library, an the alphabet of subunits may be used to connect peaks in the fragmentation spectra; each connection represents a neutral loss of an alphabet mass. This technique is called “de novo sequencing” and relies on the alphabet being known in advance. Often the alphabet of m/z difference values allowed by de novo analysis is not known or is incomplete. A method is proposed that, given fragmentation mass spectra, identifies an alphabet of m/z differences that can build large connected graphs from many intense peaks in each spectrum from a collection. Once an alphabet is obtained, it is informative to find common substructures among the peaks connected by the alphabet. This is the same as finding the largest isomorphic subgraphs on the de novo graphs from all pairs of fragmentation spectra. This maximal subgraph isomorphism problem is a generalization of the subgraph isomorphism problem, which asks whether a graph G1 has a subgraph isomorphic to a graph G2 . Subgraph isomorphism is NP-complete. A novel method of efficiently finding common substructures among the subspectra induced by the alphabet is proposed. This method is then combined with a novel form of hashing, eschewing evaluation of all pairs of fragmentation spectra. These methods are generalized to Euclidean graphs embedded in Zn

Shared-Memory Parallel Maximal Clique Enumeration

We present shared-memory parallel methods for Maximal Clique Enumeration (MCE) from a graph. MCE is a fundamental and well-studied graph analytics task, and is a widely used primitive for identifying dense structures in a graph. Due to its computationally intensive nature, parallel methods are imperative for dealing with large graphs. However, surprisingly, there do not yet exist scalable and parallel methods for MCE on a shared-memory parallel machine. In this work, we present efficient shared-memory parallel algorithms for MCE, with the following properties: (1) the parallel algorithms are provably work-efficient relative to a state-of-the-art sequential algorithm (2) the algorithms have a provably small parallel depth, showing that they can scale to a large number of processors, and (3) our implementations on a multicore machine shows a good speedup and scaling behavior with increasing number of cores, and are substantially faster than prior shared-memory parallel algorithms for MCE.Comment: 10 pages, 3 figures, proceedings of the 25th IEEE International Conference on. High Performance Computing, Data, and Analytics (HiPC), 201