Faster Algorithms for the Maximum Common Subtree Isomorphism Problem

Faster Algorithms for the Maximum Common Subtree Isomorphism Problem

Abstract

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 Delta our algorithm achieves a running time of 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 O(n^6+T*n^2) to O(n^3+T*n*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.

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Authors
  • Droschinsky, Andre
  • Kriege, Nils M.
  • Mutzel, Petra
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Shortfacts
Category
Paper in Conference Proceedings or in Workshop Proceedings (Paper)
Event Title
41st International Symposium on Mathematical Foundations of Computer Science (MFCS)
Divisions
Data Mining and Machine Learning
Event Location
Krakow, Poland
Event Type
Conference
Event Dates
22.-26.08.2016
Series Name
LIPIcs
ISSN/ISBN
978-3-95977-016-3
Publisher
Schloss Dagstuhl - Leibniz-Zentrum f\"r Informatik
Page Range
33:1-33:14
Date
22 August 2016
Official URL
https://doi.org/10.4230/LIPIcs.MFCS.2016.33
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