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

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 pol...

• Type: Conference
• Year: 2007
• Where: ISAAC
• Authors: Uffe Flarup, Pascal Koiran, Laurent Lyaudet
• Keywords: Complexity Classes Vp, Hamiltonian Polynomials, ISAAC 2007, Permanent