Faster Algorithms for the Maximum Common Subtree Isomorphism Problem

The maximum common subtree isomorphism problem asks for the largest possible isomorphism between subtrees of two given input trees. This problem is a natural restriction of the maximum common subgraph problem, which is ${\sf NP}$-hard in general graphs. Confining to trees renders polynomial time algorithms possible and is of fundamental importance for approaches on more general graph classes. Various variants of this problem in trees have been intensively studied. We consider the general case, where trees are neither rooted nor ordered and the isomorphism is maximum w.r.t. a weight function on the mapped vertices and edges. For trees of order $n$ and maximum degree $\Delta$ our algorithm achieves a running time of $\mathcal{O}(n^2\Delta)$ by exploiting the structure of the matching instances arising as subproblems. Thus our algorithm outperforms the best previously known approaches. No faster algorithm is possible for trees of bounded degree and for trees of unbounded degree we show that a further reduction of the running time would directly improve the best known approach to the assignment problem. Combining a polynomial-delay algorithm for the enumeration of all maximum common subtree isomorphisms with central ideas of our new algorithm leads to an improvement of its running time from $\mathcal{O}(n^6+Tn^2)$ to $\mathcal{O}(n^3+Tn\Delta)$, where $n$ is the order of the larger tree, $T$ is the number of different solutions, and $\Delta$ is the minimum of the maximum degrees of the input trees. Our theoretical results are supplemented by an experimental evaluation on synthetic and real-world instances

Inductive queries for a drug designing robot scientist

It is increasingly clear that machine learning algorithms need to be integrated in an iterative scientific discovery loop, in which data is queried repeatedly by means of inductive queries and where the computer provides guidance to the experiments that are being performed. In this chapter, we summarise several key challenges in achieving this integration of machine learning and data mining algorithms in methods for the discovery of Quantitative Structure Activity Relationships (QSARs). We introduce the concept of a robot scientist, in which all steps of the discovery process are automated; we discuss the representation of molecular data such that knowledge discovery tools can analyse it, and we discuss the adaptation of machine learning and data mining algorithms to guide QSAR experiments

Comparing graphs

Graphs are a well-studied mathematical concept, which has become ubiquitous to represent structured data in many application domains like computer vision, social network analysis or chem- and bioinformatics. The ever-increasing amount of data in these domains requires to efficiently organize and extract information from large graph data sets. In this context techniques for comparing graphs are fundamental, e.g., in order to obtain meaningful similarity measures between graphs. These are a prerequisite for the application of a variety of data mining algorithms to the domain of graphs. Hence, various approaches to graph comparison evolved and are wide-spread in practice. This thesis is dedicated to two different strategies for comparing graphs: maximum common subgraph problems and graph kernels. We study maximum common subgraph problems, which are based on classical graph-theoretical concepts for graph comparison and are NP-hard in the general case. We consider variants of the maximum common subgraph problem in restricted graph classes, which are highly relevant for applications in cheminformatics. We develop a polynomial-time algorithm, which allows to compute a maximum common subgraph under block and bridge preserving isomorphism in series-parallel graphs. This generalizes the problem of computing maximum common biconnected subgraphs in series-parallel graphs. We show that previous approaches to this problem, which are based on the separators represented by standard graph decompositions, fail. We introduce the concept of potential separators to overcome this issue and use them algorithmically to solve the problem in series-parallel graphs. We present algorithms with improved bounds on running time for the subclass of outerplanar graphs. Finally, we establish a sufficient condition for maximum common subgraph variants to allow derivation of graph distance metrics. This leads to polynomial-time computable graph distance metrics in restricted graph classes. This progress constitutes a step towards solving practically relevant maximum common subgraph problems in polynomial time. The second contribution of this thesis is to graph kernels, which have their origin in specific data mining algorithms. A key property of graph kernels is that they allow to consider a large (possibly infinite) number of features and can support graphs with arbitrary annotation, while being efficiently computable. The main contributions of this part of the thesis are (i) the development of novel graph kernels, which are especially designed for attributed graphs with arbitrary annotations and (ii) the systematic study of implicit and explicit mapping into a feature space for computation of graph kernels w.r.t. its impact on the running time and the ability to consider arbitrary annotations. We propose graph kernels based on bijections between subgraphs and walks of fixed length. In an experimental study we show that these approaches provide a viable alternative to known techniques, in particular for graphs with complex annotations

A Survey on Graph Kernels

Graph kernels have become an established and widely-used technique for solving classification tasks on graphs. This survey gives a comprehensive overview of techniques for kernel-based graph classification developed in the past 15 years. We describe and categorize graph kernels based on properties inherent to their design, such as the nature of their extracted graph features, their method of computation and their applicability to problems in practice. In an extensive experimental evaluation, we study the classification accuracy of a large suite of graph kernels on established benchmarks as well as new datasets. We compare the performance of popular kernels with several baseline methods and study the effect of applying a Gaussian RBF kernel to the metric induced by a graph kernel. In doing so, we find that simple baselines become competitive after this transformation on some datasets. Moreover, we study the extent to which existing graph kernels agree in their predictions (and prediction errors) and obtain a data-driven categorization of kernels as result. Finally, based on our experimental results, we derive a practitioner's guide to kernel-based graph classification

Efficient Frequent Subtree Mining Beyond Forests

A common paradigm in distance-based learning is to embed the instance space into some appropriately chosen feature space equipped with a metric and to define the dissimilarity between instances by the distance of their images in the feature space. If the instances are graphs, then frequent connected subgraphs are a well-suited pattern language to define such feature spaces. Identifying the set of frequent connected subgraphs and subsequently computing embeddings for graph instances, however, is computationally intractable. As a result, existing frequent subgraph mining algorithms either restrict the structural complexity of the instance graphs or require exponential delay between the output of subsequent patterns. Hence distance-based learners lack an efficient way to operate on arbitrary graph data. To resolve this problem, in this thesis we present a mining system that gives up the demand on the completeness of the pattern set to instead guarantee a polynomial delay between subsequent patterns. Complementing this, we devise efficient methods to compute the embedding of arbitrary graphs into the Hamming space spanned by our pattern set. As a result, we present a system that allows to efficiently apply distance-based learning methods to arbitrary graph databases. To overcome the computational intractability of the mining step, we consider only frequent subtrees for arbitrary graph databases. This restriction alone, however, does not suffice to make the problem tractable. We reduce the mining problem from arbitrary graphs to forests by replacing each graph by a polynomially sized forest obtained from a random sample of its spanning trees. This results in an incomplete mining algorithm. However, we prove that the probability of missing a frequent subtree pattern is low. We show empirically that this is true in practice even for very small sized forests. As a result, our algorithm is able to mine frequent subtrees in a range of graph databases where state-of-the-art exact frequent subgraph mining systems fail to produce patterns in reasonable time or even at all. Furthermore, the predictive performance of our patterns is comparable to that of exact frequent connected subgraphs, where available. The above method considers polynomially many spanning trees for the forest, while many graphs have exponentially many spanning trees. The number of patterns found by our mining algorithm can be negatively influenced by this exponential gap. We hence propose a method that can (implicitly) consider forests of exponential size, while remaining computationally tractable. This results in a higher recall for our incomplete mining algorithm. Furthermore, the methods extend the known positive results on the tractability of exact frequent subtree mining to a novel class of transaction graphs. We conjecture that the next natural extension of our results to a larger transaction graph class is at least as difficult as proving whether P = NP, or not. Regarding the graph embedding step, we apply a similar strategy as in the mining step. We represent a novel graph by a forest of its spanning trees and decide whether the frequent trees from the mining step are subgraph isomorphic to this forest. As a result, the embedding computation has one-sided error with respect to the exact subgraph isomorphism test but is computationally tractable. Furthermore, we show that we can leverage a partial order on the pattern set. This structure can be used to reduce the runtime of the embedding computation dramatically. For the special case of Jaccard-similarity between graph embeddings, a further substantial reduction of runtime can be achieved using min-hashing. The Jaccard-distance can be approximated using small sketch vectors that can be computed fast, again using the partial order on the tree patterns

FS^3: A Sampling based method for top-k Frequent Subgraph Mining

Mining labeled subgraph is a popular research task in data mining because of its potential application in many different scientific domains. All the existing methods for this task explicitly or implicitly solve the subgraph isomorphism task which is computationally expensive, so they suffer from the lack of scalability problem when the graphs in the input database are large. In this work, we propose FS^3, which is a sampling based method. It mines a small collection of subgraphs that are most frequent in the probabilistic sense. FS^3 performs a Markov Chain Monte Carlo (MCMC) sampling over the space of a fixed-size subgraphs such that the potentially frequent subgraphs are sampled more often. Besides, FS^3 is equipped with an innovative queue manager. It stores the sampled subgraph in a finite queue over the course of mining in such a manner that the top-k positions in the queue contain the most frequent subgraphs. Our experiments on database of large graphs show that FS^3 is efficient, and it obtains subgraphs that are the most frequent amongst the subgraphs of a given size

Kernel Functions for Graph Classification

Graphs are information-rich structures, but their complexity makes them difficult to analyze. Given their broad and powerful representation capacity, the classification of graphs has become an intense area of research. Many established classifiers represent objects with vectors of explicit features. When the number of features grows, however, these vector representations suffer from typical problems of high dimensionality such as overfitting and high computation time. This work instead focuses on using kernel functions to map graphs into implicity defined spaces that avoid the difficulties of vector representations. The introduction of kernel classifiers has kindled great interest in kernel functions for graph data. By using kernels the problem of graph classification changes from finding a good classifier to finding a good kernel function. This work explores several novel uses of kernel functions for graph classification. The first technique is the use of structure based features to add structural information to the kernel function. A strength of this approach is the ability to identify specific structure features that contribute significantly to the classification process. Discriminative structures can then be passed off to domain-specific researchers for additional analysis. The next approach is the use of wavelet functions to represent graph topology as simple real-valued features. This approach achieves order-of-magnitude decreases in kernel computation time by eliminating costly topological comparisons, while retaining competitive classification accuracy. Finally, this work examines the use of even simpler graph representations and their utility for classification. The models produced from the kernel functions presented here yield excellent performance with respect to both efficiency and accuracy, as demonstrated in a variety of experimental studies