On the Expressive Power of Permanents and Perfect Matchings of Matrices of Bounded Pathwidth/Cliquewidth (Extended Abstract)
Some 25 years ago Valiant introduced an algebraic model of computation in order to study the complexity of evaluating families of polynomials. The theory was introduced along with the complexity classes Open image in new window and Open image in new window which are analogues of the classical classes Open image in new window and Open image in new window . Families of polynomials that are difficult to evaluate (that is, Open image in new window -complete) includes the permanent and hamiltonian polynomials.
In a previous paper the authors together with P. Koiran studied the expressive power of permanent and hamiltonian polynomials of matrices of bounded treewidth, as well as the expressive power of perfect matchings of planar graphs. It was established that the permanent and hamiltonian polynomials of matrices of bounded treewidth are equivalent to arithmetic formulas. Also, the sum of weights of perfect matchings of planar graphs was shown to be equivalent to (weakly) skew circuits.
In this paper we continue the research in the direction described above, and study the expressive power of permanents, hamiltonians and perfect matchings of matrices that have bounded pathwidth or bounded cliquewidth. In particular, we prove that permanents, hamiltonians and perfect matchings of matrices that have bounded pathwidth express exactly arithmetic formulas. This is an improvement of our previous result for matrices of bounded treewidth. Also, for matrices of bounded weighted cliquewidth we show membership in Open image in new window for these polynomials.
KeywordsPlanar Graph Weighted Graph Hamiltonian Cycle Expressive Power Parse Tree
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