On Maximum Common Subgraph Problems in Series-Parallel Graphs

Kriege, Nils M.; Kurpicz, Florian; Mutzel, Petra

doi:10.1007/978-3-319-19315-1_18

Cited by 6 publications

(10 citation statements)

References 20 publications

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“…A common principle in comparing graphs is identifying correspondences between their nodes that optimally preserve the edge structure. The problem arises in various domains and variations have been studied under di erent terms such as maximum common subgraph isomorphism [114], network alignment [220], graph matching [72] or graph edit distance [178], often using di erent algorithmic approaches. These problems are NP-hard, and in practice, heuristics are widely applied.…”

Section: Graph Matchingmentioning

confidence: 99%

Weisfeiler and Leman go Machine Learning: The Story so far

Morris¹,

Lipman²,

Maron³

et al. 2021

Preprint

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In recent years, algorithms and neural architectures based on the Weisfeiler-Leman algorithm, a well-known heuristic for the graph isomorphism problem, emerged as a powerful tool for machine learning with graphs and relational data. Here, we give a comprehensive overview of the algorithm's use in a machine learning setting, focusing on the supervised regime. We discuss the theoretical background, show how to use it for supervised graph-and node representation learning, discuss recent extensions, and outline the algorithm's connection to (permutation-)equivariant neural architectures. Moreover, we give an overview of current applications and future directions to stimulate further research.

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Section: Graph Matchingmentioning

confidence: 99%

Weisfeiler and Leman go Machine Learning: The Story so far

Morris¹,

Lipman²,

Maron³

et al. 2021

Preprint

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show abstract

“…We assume that this approach is realized by the procedure MaximalIso(uv, u v , t), 7 return maximum entry in D which returns the unique maximal isomorphism that maps the two given edges according to the specified type. The algorithm can be implemented by means of a tree structure that encodes the neighboring relation between inner faces, e.g., SP-trees as in [6,7] or weak dual graphs similar to the approach of [13].…”

Section: Biconnected Mcis In Outerplanar Graphsmentioning

confidence: 99%

“…BBP-MCES is not only computable in polynomial-time, but also yields meaningful results for cheminformatics. A polynomial-time algorithm was recently proposed for BBP-MCIS, which requires time O(n 6 ) in series-parallel and O(n 5 ) in outerplanar graphs [6].…”

Section: Introductionmentioning

confidence: 99%

Finding Largest Common Substructures of Molecules in Quadratic Time

Droschinsky

Kriege

Mutzel

2017

SOFSEM 2017: Theory and Practice of Computer Science

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Finding the common structural features of two molecules is a fundamental task in cheminformatics. Most drugs are small molecules, which can naturally be interpreted as graphs. Hence, the task is formalized as maximum common subgraph problem. Albeit the vast majority of molecules yields outerplanar graphs this problem remains NP-hard. We consider a variation of the problem of high practical relevance, where the rings of molecules must not be broken, i.e., the block and bridge structure of the input graphs must be retained by the common subgraph. We present an algorithm for finding a maximum common connected induced subgraph of two given outerplanar graphs subject to this constraint. Our approach runs in time O(∆n 2 ) in outerplanar graphs on n vertices with maximum degree ∆. This leads to a quadratic time complexity in molecular graphs, which have bounded degree. The experimental comparison on synthetic and real-world datasets shows that our approach is highly efficient in practice and outperforms comparable state-of-the-art algorithms.

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“…If two such components share a vertex, the corresponding B-vertices are connected through a C-vertex representing this shared vertex. The running time for MCST on BC-trees is an important factor for the total running time of MCS algorithms for outerplanar and series-parallel graphs like [2,13,18] compared to FOG's 481.3 ms. The speedup factor ranges from 24 to 59, with an average of 43.…”

Section: Experimental Comparisonmentioning

confidence: 99%

“…This problem is directly relevant in various applications, where real-world objects like molecules or shapes are represented by (attributed) trees [16,20]. Moreover, it forms the basis for several recent approaches to solve MCS in more general graph classes, see [2,13,14,18]. Methods directly based on algorithms for rooted trees result in time O(n 4 ∆) by considering all pairs of possible roots.…”

Section: Introductionmentioning

confidence: 99%

Faster Algorithms for the Maximum Common Subtree Isomorphism Problem

Droschinsky,

Kriege,

Mutzel

2016

Preprint

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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 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 ∆ our algorithm achieves a running time of O(n 2 ∆) 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 polynomialdelay 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 O(n 6 + T n 2 ) to O(n 3 + T n∆), where n is the order of the larger tree, T 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.

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On Maximum Common Subgraph Problems in Series-Parallel Graphs

Cited by 6 publications

References 20 publications

Weisfeiler and Leman go Machine Learning: The Story so far

Weisfeiler and Leman go Machine Learning: The Story so far

Finding Largest Common Substructures of Molecules in Quadratic Time

Faster Algorithms for the Maximum Common Subtree Isomorphism Problem

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