Capturing Topology in Graph Pattern Matching

Graph pattern matching is often defined in terms of subgraph isomorphism, an NP-complete problem. To lower its complexity, various extensions of graph simulation have been considered instead. These extensions allow pattern matching to be conducted in cubic-time. However, they fall short of capturing the topology of data graphs, i.e., graphs may have a structure drastically different from pattern graphs they match, and the matches found are often too large to understand and analyze. To rectify these problems, this paper proposes a notion of strong simulation, a revision of graph simulation, for graph pattern matching. (1) We identify a set of criteria for preserving the topology of graphs matched. We show that strong simulation preserves the topology of data graphs and finds a bounded number of matches. (2) We show that strong simulation retains the same complexity as earlier extensions of simulation, by providing a cubic-time algorithm for computing strong simulation. (3) We present the locality property of strong simulation, which allows us to effectively conduct pattern matching on distributed graphs. (4) We experimentally verify the effectiveness and efficiency of these algorithms, using real-life data and synthetic data.Comment: VLDB201

A Logic that Captures β\betaP on Ordered Structures

We extend the inflationary fixed-point logic, IFP, with a new kind of second-order quantifiers which have (poly-)logarithmic bounds. We prove that on ordered structures the new logic $\exists^{\log^{\omega}}\text{IFP}$ captures the limited nondeterminism class $\beta\text{P}$. In order to study its expressive power, we also design a new version of Ehrenfeucht-Fra\"iss\'e game for this logic and show that our capturing result will not hold on the general case, i.e. on all the finite structures.Comment: 15 pages. This article was reported with a title "Logarithmic-Bounded Second-Order Quantifiers and Limited Nondeterminism" in National Conference on Modern Logic 2019, on November 9 in Beijin