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Valiant introduced some 25 years ago an algebraic model of computation along with the complexity classes VP and VNP, which can be viewed as analogues of the classical classes P and NP. Prominent examples of difficult (that is, VNP-complete) problems in this model includes the permanent and hamiltonian polynomials. In this paper we investigate the expressive power of easy special cases of these polynomials. We show that the permanent and hamiltonian polynomials for matrices of bounded treewidth both are equivalent to arithmetic formulas. Also, arithmetic weakly skew circuits are shown to be equivalent to the sum of weights of perfect matchings of planar graphs.

On the Expressive Power of Planar Perfect Matching and Permanents of Bounded Treewidth Matrices