20 research outputs found
A Theory for Valiant's Matchcircuits (Extended Abstract)
The computational function of a matchgate is represented by its character
matrix. In this article, we show that all nonsingular character matrices are
closed under matrix inverse operation, so that for every , the nonsingular
character matrices of -bit matchgates form a group, extending the recent
work of Cai and Choudhary (2006) of the same result for the case of , and
that the single and the two-bit matchgates are universal for matchcircuits,
answering a question of Valiant (2002)
Counting degree-constrained subgraphs and orientations
The goal of this short paper to advertise the method of gauge transformations
(aka holographic reduction, reparametrization) that is well-known in
statistical physics and computer science, but less known in combinatorics. As
an application of it we give a new proof of a theorem of A. Schrijver asserting
that the number of Eulerian orientations of a --regular graph on
vertices with even is at least
. We also show that a
--regular graph with even has always at least as many Eulerian
orientations as --regular subgraphs
Normal Factor Graphs and Holographic Transformations
This paper stands at the intersection of two distinct lines of research. One
line is "holographic algorithms," a powerful approach introduced by Valiant for
solving various counting problems in computer science; the other is "normal
factor graphs," an elegant framework proposed by Forney for representing codes
defined on graphs. We introduce the notion of holographic transformations for
normal factor graphs, and establish a very general theorem, called the
generalized Holant theorem, which relates a normal factor graph to its
holographic transformation. We show that the generalized Holant theorem on the
one hand underlies the principle of holographic algorithms, and on the other
hand reduces to a general duality theorem for normal factor graphs, a special
case of which was first proved by Forney. In the course of our development, we
formalize a new semantics for normal factor graphs, which highlights various
linear algebraic properties that potentially enable the use of normal factor
graphs as a linear algebraic tool.Comment: To appear IEEE Trans. Inform. Theor
P versus NP and geometry
I describe three geometric approaches to resolving variants of P v. NP,
present several results that illustrate the role of group actions in complexity
theory, and make a first step towards completely geometric definitions of
complexity classes.Comment: 20 pages, to appear in special issue of J. Symbolic. Comp. dedicated
to MEGA 200
A Recursive Definition of the Holographic Standard Signature
We provide a recursive description of the signatures realizable on the
standard basis by a holographic algorithm. The description allows us to prove
tight bounds on the size of planar matchgates and efficiently test for standard
signatures. Over finite fields, it allows us to count the number of n-bit
standard signatures and calculate their expected sparsity.Comment: Fixed small typo in Section 3.
Local Statistics of Realizable Vertex Models
We study planar "vertex" models, which are probability measures on edge
subsets of a planar graph, satisfying certain constraints at each vertex,
examples including dimer model, and 1-2 model, which we will define. We express
the local statistics of a large class of vertex models on a finite hexagonal
lattice as a linear combination of the local statistics of dimers on the
corresponding Fisher graph, with the help of a generalized holographic
algorithm. Using an torus to approximate the periodic infinite
graph, we give an explicit integral formula for the free energy and local
statistics for configurations of the vertex model on an infinite bi-periodic
graph. As an example, we simulate the 1-2 model by the technique of Glauber
dynamics