On the expressive power of planar perfect matching and permanents of bounded treewidth matrices

TL;DR

This paper explores the computational complexity of permanents and Hamiltonian polynomials for matrices with bounded treewidth, revealing their equivalence to arithmetic formulas and linking perfect matchings in planar graphs to weakly skew circuits.

Contribution

It establishes the equivalence of permanents and Hamiltonian polynomials for bounded treewidth matrices to arithmetic formulas, and connects perfect matchings in planar graphs to weakly skew circuits.

Findings

Permanents and Hamiltonian polynomials for bounded treewidth matrices are equivalent to arithmetic formulas.

Sum of weights of perfect matchings in planar graphs corresponds to arithmetic weakly skew circuits.

Results unify complexity classes for special cases of VNP-complete problems.

Abstract

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. They are defined using non-uniform sequences of arithmetic circuits and provides a framework to study the complexity for sequences of polynomials. Prominent examples of difficult (that is, VNP-complete) problems in this model includes the permanent and hamiltonian polynomials. While the permanent and hamiltonian polynomials in general are difficult to evaluate, there have been research on which special cases of these polynomials admits efficient evaluation. For instance, Barvinok has shown that if the underlying matrix has bounded rank, both the permanent and the hamiltonian polynomials can be evaluated in polynomial time, and thus are in VP. Courcelle, Makowsky and Rotics have shown that for matrices of bounded treewidth several difficult problems (including evaluating the permanent and hamiltonian polynomials) can be solved efficiently. An earlier result of this flavour is Kasteleyn's theorem which states that the sum of weights of perfect matchings of a planar graph can be computed in polynomial time, and thus is in VP also. For general graphs this problem is VNP-complete. In this paper we investigate the expressive power of the above results. 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.