7,023 research outputs found
Between Subgraph Isomorphism and Maximum Common Subgraph
When a small pattern graph does not occur inside a larger target graph, we can ask how to find "as much of the pattern as possible" inside the target graph. In general, this is known as the maximum common subgraph problem, which is much more computationally challenging in practice than subgraph isomorphism. We introduce a restricted alternative, where we ask if all but k vertices from the pattern can be found in the target graph. This allows for the development of slightly weakened forms of certain invariants from subgraph isomorphism which are based upon degree and number of paths. We show that when k is small, weakening the invariants still retains much of their effectiveness. We are then able to solve this problem on the standard problem instances used to benchmark subgraph isomorphism algorithms, despite these instances being too large for current maximum common subgraph algorithms to handle. Finally, by iteratively increasing k, we obtain an algorithm which is also competitive for the maximum common subgraph
Graph theoretic methods for the analysis of structural relationships in biological macromolecules
Subgraph isomorphism and maximum common subgraph isomorphism algorithms from graph theory provide an effective and an efficient way of identifying structural relationships between biological macromolecules. They thus provide a natural complement to the pattern matching algorithms that are used in bioinformatics to identify sequence relationships. Examples are provided of the use of graph theory to analyze proteins for which three-dimensional crystallographic or NMR structures are available, focusing on the use of the Bron-Kerbosch clique detection algorithm to identify common folding motifs and of the Ullmann subgraph isomorphism algorithm to identify patterns of amino acid residues. Our methods are also applicable to other types of biological macromolecule, such as carbohydrate and nucleic acid structures
Effectiveness of graph-based and fingerprint-based similarity measures for virtual screening of 2D chemical structure databases
This paper reports an evaluation of both graph-based and fingerprint-based measures of structural similarity, when used for virtual screening of sets of 2D molecules drawn from the MDDR and ID Alert databases. The graph-based measures employ a new maximum common edge subgraph isomorphism algorithm, called RASCAL, with several similarity coefficients described previously for quantifying the similarity between pairs of graphs. The effectiveness of these graph-based searches is compared with that resulting from similarity searches using BCI, Daylight and Unity 2D fingerprints. Our results suggest that graph-based approaches provide an effective complement to existing fingerprint-based approaches to virtual screening
A Partitioning Algorithm for Maximum Common Subgraph Problems
We introduce a new branch and bound algorithm for the maximum common subgraph and maximum common connected subgraph problems which is based around vertex labelling and partitioning. Our method in some ways resembles a traditional constraint programming approach, but uses a novel compact domain store and supporting inference algorithms which dramatically reduce the memory and computation requirements during search, and allow better dual viewpoint ordering heuristics to be calculated cheaply. Experiments show a speedup of more than an order of magnitude over the state of the art, and demonstrate that we can operate on much larger graphs without running out of memory
Maximum common subgraph isomorphism algorithms for the matching of chemical structures
The maximum common subgraph (MCS) problem has become increasingly important in those aspects of chemoinformatics that involve the matching of 2D or 3D chemical structures. This paper provides a classification and a review of the many MCS algorithms, both exact and approximate, that have been described in the literature, and makes recommendations regarding their applicability to typical chemoinformatics tasks
A Necessary and Sufficient Condition for Graph Matching to be equivalent to Clique Search
This paper formulates a necessary and sufficient condition for a generic graph matching problem to be equivalent to the maximum vertex and edge weight clique problem in a derived association graph. The consequences of this results are threefold: first, the condition is general enough to cover a broad range of practical graph matching problems; second, a proof to establish equivalence between graph matching and clique search reduces to showing that a given graph matching problem satisfies the proposed condition;\ud
and third, the result sets the scene for generic continuous solutions for a broad range of graph matching problems. To illustrate the mathematical framework, we apply it to a number of graph matching problems, including the problem of determining the graph edit distance
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 -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 and maximum degree
our algorithm achieves a running time of 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 to ,
where is the order of the larger tree, is the number of different
solutions, and 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
Subgraph Matching Kernels for Attributed Graphs
We propose graph kernels based on subgraph matchings, i.e.
structure-preserving bijections between subgraphs. While recently proposed
kernels based on common subgraphs (Wale et al., 2008; Shervashidze et al.,
2009) in general can not be applied to attributed graphs, our approach allows
to rate mappings of subgraphs by a flexible scoring scheme comparing vertex and
edge attributes by kernels. We show that subgraph matching kernels generalize
several known kernels. To compute the kernel we propose a graph-theoretical
algorithm inspired by a classical relation between common subgraphs of two
graphs and cliques in their product graph observed by Levi (1973). Encouraging
experimental results on a classification task of real-world graphs are
presented.Comment: Appears in Proceedings of the 29th International Conference on
Machine Learning (ICML 2012
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