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Tree width measures how close a graph is to a tree. It is NP-hard to compute tree width. The best known approximation algorithm achieves $O(\sqrt{{\log}n})$ factor.

Courcelle's theorem states that any property of graphs definable in monadic second-order logic (MSO2) can be decided in linear time on any class of graphs of bounded tree width. A recent paper showed that Courcelle's theorem still holds when "linear time" is replaced with "logspace". However, this does not settle the space complexity of Graph Isomorphism on graphs with bounded tree width. The best known result puts it in LogCFL.

Are there other problems that are :

  • NP-hard (or not known to be in P) on general graphs, and
  • known to be solvable in linear/polynomial time on graphs with bounded tree width, and
  • NOT known to be in LogSpace ?
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  • $\begingroup$ What is the $n$ mentioned in the approximation factor? $\endgroup$ – gphilip Sep 16 '10 at 2:37
  • $\begingroup$ $n$ is the number of vertices in the input graph. $\endgroup$ – Shiva Kintali Sep 16 '10 at 3:03
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    $\begingroup$ We can, in general, do better than $O(\sqrt{\log n})$ in approximating treewidth. The best polynomial-time approximation algorithm that I am aware of achieves an $O(\sqrt{\log w})$ approximation factor, where $w$ is the treewidth of the graph. See Feige, Hajiaghayi, and Lee, Improved approximation algorithms for minimum-weight vertex separators , STOC 2005. $\endgroup$ – gphilip Sep 16 '10 at 14:06
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Tutte polynomial is an example.

This is a generalization of the chromatic polynomial, which itself is a #P-hard problem in any reasonable formulation. In

Evaluating the Tutte Polynomial for Graphs of Bounded Tree-Width, S. D. Noble, Combinatorics, Probability and Computing, 1998,

a polynomial time algorithm is given where the time complexity is about $O(f(k)n^{4+\epsilon})$, where $k$ is the treewidth and $n$ is the number of nodes. Related works can be seen in here. For a survey, there are article on Arxiv, where the complexity issue discussed in section 8.

It seems that the problem cannot be expressed in MSO2 directly, though I'm not familiar with the detailed definitions... Hope this problem is what you need!

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  • $\begingroup$ What is the function f? $\endgroup$ – Michael Blondin Oct 20 '10 at 3:02
  • $\begingroup$ According to the paper, it is a function in $k$ with order about $O(2^{(k^{3+\epsilon})})$. $\endgroup$ – Hsien-Chih Chang 張顯之 Oct 20 '10 at 3:12
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    $\begingroup$ Makowski says that "all the graph polynomials studied in the literature are SOL-definable", and gives a formulation of Tutte polynomial in terms of SOL, page 15 of "From a Zoo to a Zoology: Towards a General Theory of Graph Polynomials." In "Algorithmic uses of the Feferman-Vaught theorem" he extends Courcelle's theorem to show tractability for SOL-definable polynomials on bounded tree-width graphs. $\endgroup$ – Yaroslav Bulatov Oct 21 '10 at 23:05

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