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International Conference on Theory and Applications of Models of Computation

TAMC 2015: Theory and Applications of Models of Computation pp 297–308Cite as

Bounded Treewidth and Space-Efficient Linear Algebra

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9076))

Abstract

Motivated by a recent result of Elberfeld, Jakoby and Tantau [EJT10] showing that \(\mathsf {MSO}\) properties are Logspace computable on graphs of bounded treewidth, we consider the complexity of computing the determinant of the adjacency matrix of a bounded treewidth graph and as our main result prove that it is in Logspace. It is important to notice that the determinant is neither an \(\mathsf {MSO}\)-property nor counts the number of solutions of an \(\mathsf {MSO}\)-predicate. This technique yields Logspace algorithms for counting the number of spanning arborescences and directed Euler tours in bounded treewidth digraphs.

We demonstrate some linear algebraic applications of the determinant algorithm by describing Logspace procedures for the characteristic polynomial, the powers of a weighted bounded treewidth graph and feasibility of a system of linear equations where the underlying bipartite graph has bounded treewidth.

Finally, we complement our upper bounds by proving L-hardness of the problems of computing the determinant, and of powering a bounded treewidth matrix. We also show the GapL-hardness of Iterated Matrix Multiplication where each matrix has bounded treewidth.

S. Datta—Part of the work was done on a visit to the Institute for Theoretical Computer Science at Leibniz University Hannover.

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Notes

  1. 1.

    The case when quantification over subset of edges is not allowed is referred to as \(\mathsf {MSO}_1\) which is known to be strictly less powerful than \(\mathsf {MSO}_2\), the case when edge set quantification is allowed. Throughtout our paper, we will work with \(\mathsf {MSO}_2\) and hence we will just refer to it as \(\mathsf {MSO}\).

  2. 2.

    The Gaifman graph (also called the Primal Graph) of a binary relation \(R \subseteq A\,\times \,A\) is the graph whose nodes are elements of \(A\) and an edge joins a pair of variables \(x,y\) if \((x,y) \in R\).

  3. 3.

    Binary relation \(R\) is said to be compatible with the tree decomposition \(T'\) of \(G\) if the Gaifman graph of \(R\) has \(T'\) as its tree decomposition.

  4. 4.

    Note that since we require that for a given \(X,Y\), every \(v \in V\) has a unique \(h \in X\), our formula is not monotone, i.e., If \(X \subseteq X'\) are two sets of heads then if \(\phi (X,Y)\) is true doesn’t imply \(\phi (X',Y)\) is also true (consider vertices in \(X'\setminus X\), since \(X' \subseteq X\), they will have two different \(h, h'\) such that the \(\mathsf {PATH}\) and \({\mathsf {NXT}}^{*}\) predicates are true contradicting uniqueness of \(h\).

  5. 5.

    Alternately, arborescences are \(\mathsf {MSO}_2\)-definable and, thus, counting them in bounded treewidth graphs can be implemented in L via [EJT10].

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Acknowledgement

We would like to thank Abhishek Bhrushundi, Arne Meier, Rohith Varma and Heribert Vollmer for illuminating discussions regarding this paper. Special thanks are due to Johannes Köbler and Sebastian Kuhnert who were involved in the initial discussions on the proof of Theorems 4, 6; to Stefan Mengel for proof reading the paper and finding a gap in a previous “proof” of Theorem 4; and to Raghav Kulkarni for suggesting proof strategies for Corollary 5 and Lemma 3; and to Sebastian Kuhnert for the proof of Proposition 2. Thanks are also due to anonymous referees for pointing out errors in a previous version of the paper and for greatly simplifying the proof of Corollary 3. This work is partially funded by a grant from Infosys Foundation.

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Authors and Affiliations

  1. Chennai Mathematical Institute (CMI), Chennai, India

    Nikhil Balaji & Samir Datta

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  1. Nikhil Balaji
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  2. Samir Datta
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Correspondence to Nikhil Balaji .

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Editors and Affiliations

  1. National University of Singapore, Singapore, Singapore

    Rahul Jain

  2. National University of Singapore, Singapore, Singapore

    Sanjay Jain

  3. National University of Singapore, Singapore, Singapore

    Frank Stephan

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Balaji, N., Datta, S. (2015). Bounded Treewidth and Space-Efficient Linear Algebra. In: Jain, R., Jain, S., Stephan, F. (eds) Theory and Applications of Models of Computation. TAMC 2015. Lecture Notes in Computer Science(), vol 9076. Springer, Cham. https://doi.org/10.1007/978-3-319-17142-5_26

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  • DOI: https://doi.org/10.1007/978-3-319-17142-5_26

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