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Log-Space Algorithms for Paths and Matchings in k-Trees

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Reachability and shortest path problems are NL-complete for general graphs. They are known to be in L for graphs of tree-width 2 (Jakoby and Tantau in Proceedings of FSTTCS’07: The 27th Annual Conference on Foundations of Software Technology and Theoretical Computer Science, pp. 216–227, 2007). In this paper, we improve these bounds for k-trees, where k is a constant. In particular, the main results of our paper are log-space algorithms for reachability in directed k-trees, and for computation of shortest and longest paths in directed acyclic k-trees.

Besides the path problems mentioned above, we also consider the problem of deciding whether a k-tree has a perfect matching (decision version), and if so, finding a perfect matching (search version), and prove that these two problems are L-complete. These problems are known to be in P and in RNC for general graphs, and in SPL for planar bipartite graphs, as shown in Datta et al. (Theory Comput. Syst. 47:737–757, 2010).

Our results settle the complexity of these problems for the class of k-trees. The results are also applicable for bounded tree-width graphs, when a tree-decomposition is given as input. The technique central to our algorithms is a careful implementation of the divide-and-conquer approach in log-space, along with some ideas from Jakoby and Tantau (Proceedings of FSTTCS’07: The 27th Annual Conference on Foundations of Software Technology and Theoretical Computer Science, pp. 216–227, 2007) and Limaye et al. (Theory Comput. Syst. 46(3):499–522, 2010).

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Authors and Affiliations

  1. Indian Institute of Technology, Gandhinagar, India

    Bireswar Das

  2. Chennai Mathematical Institute, Chennai, India

    Samir Datta & Prajakta Nimbhorkar

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  1. Bireswar Das
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  2. Samir Datta
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  3. Prajakta Nimbhorkar
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Correspondence to Prajakta Nimbhorkar.

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Das, B., Datta, S. & Nimbhorkar, P. Log-Space Algorithms for Paths and Matchings in k-Trees. Theory Comput Syst 53, 669–689 (2013). https://doi.org/10.1007/s00224-013-9469-9

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