12,468 research outputs found
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 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.Comment: 14 page
Evaluating a weighted graph polynomial for graphs of bounded tree-width
We show that for any there is a polynomial time algorithm to evaluate the weighted graph polynomial of any graph with tree-width at most at any point. For a graph with vertices, the algorithm requires arithmetical operations, where depends only on
Dimers and cluster integrable systems
We show that the dimer model on a bipartite graph on a torus gives rise to a
quantum integrable system of special type - a cluster integrable system. The
phase space of the classical system contains, as an open dense subset, the
moduli space of line bundles with connections on the graph. The sum of
Hamiltonians is essentially the partition function of the dimer model. Any
graph on a torus gives rise to a bipartite graph on the torus. We show that the
phase space of the latter has a Lagrangian subvariety. We identify it with the
space parametrizing resistor networks on the original graph.We construct
several discrete quantum integrable systems.Comment: This is an updated version, 75 pages, which will appear in Ann. Sci.
EN
On Weakly Distinguishing Graph Polynomials
A univariate graph polynomial P(G;X) is weakly distinguishing if for almost
all finite graphs G there is a finite graph H with P(G;X)=P(H;X). We show that
the clique polynomial and the independence polynomial are weakly
distinguishing. Furthermore, we show that generating functions of induced
subgraphs with property C are weakly distinguishing provided that C is of
bounded degeneracy or tree-width. The same holds for the harmonious chromatic
polynomial
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