46 research outputs found
Determinants and Perfect Matchings
We give a combinatorial interpretation of the determinant of a matrix as a
generating function over Brauer diagrams in two different but related ways. The
sign of a permutation associated to its number of inversions in the Leibniz
formula for the determinant is replaced by the number of crossings in the
Brauer diagram. This interpretation naturally explains why the determinant of
an even antisymmetric matrix is the square of a Pfaffian.Comment: 15 pages, terminology improved, exposition tightened, "deranged
matchings" example remove
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
A new approach to solving three combinatorial enumeration problems on planar graphs
The purpose of this paper is to show how the technique of delta-wye graph reduction provides an alternative method for solving three enumerative function evaluation problems on planar graphs. In particular, it is shown how to compute the number of spanning trees and perfect matchings, and how to evaluate energy in the Ising spin glass model of statistical mechanics. These alternative algorithms require O(n2) arithmetic operations on an n-vertex planar graph, and are relatively easy to implement
Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without fixed-trace
The degree of entanglement of random pure states in bipartite quantum systems
can be estimated from the distribution of the extreme Schmidt eigenvalues. For
a bipartition of size M\geq N, these are distributed according to a
Wishart-Laguerre ensemble (WL) of random matrices of size N x M, with a
fixed-trace constraint. We first compute the distribution and moments of the
smallest eigenvalue in the fixed trace orthogonal WL ensemble for arbitrary
M\geq N. Our method is based on a Laplace inversion of the recursive results
for the corresponding orthogonal WL ensemble by Edelman. Explicit examples are
given for fixed N and M, generalizing and simplifying earlier results. In the
microscopic large-N limit with M-N fixed, the orthogonal and unitary WL
distributions exhibit universality after a suitable rescaling and are therefore
independent of the constraint. We prove that very recent results given in terms
of hypergeometric functions of matrix argument are equivalent to more explicit
expressions in terms of a Pfaffian or determinant of Bessel functions. While
the latter were mostly known from the random matrix literature on the QCD Dirac
operator spectrum, we also derive some new results in the orthogonal symmetry
class.Comment: 25 pag., 4 fig - minor changes, typos fixed. To appear in JSTA
Parallel Polynomial Permanent Mod Powers of 2 and Shortest Disjoint Cycles
We present a parallel algorithm for permanent mod 2^k of a matrix of
univariate integer polynomials. It places the problem in ParityL subset of
NC^2. This extends the techniques of [Valiant], [Braverman, Kulkarni, Roy] and
[Bj\"orklund, Husfeldt], and yields a (randomized) parallel algorithm for
shortest 2-disjoint paths improving upon the recent result from (randomized)
polynomial time.
We also recognize the disjoint paths problem as a special case of finding
disjoint cycles, and present (randomized) parallel algorithms for finding a
shortest cycle and shortest 2-disjoint cycles passing through any given fixed
number of vertices or edges