66 research outputs found
Computing the partition function for graph homomorphisms
We introduce the partition function of edge-colored graph homomorphisms, of
which the usual partition function of graph homomorphisms is a specialization,
and present an efficient algorithm to approximate it in a certain domain.
Corollaries include efficient algorithms for computing weighted sums
approximating the number of k-colorings and the number of independent sets in a
graph, as well as an efficient procedure to distinguish pairs of edge-colored
graphs with many color-preserving homomorphisms G --> H from pairs of graphs
that need to be substantially modified to acquire a color-preserving
homomorphism G --> H.Comment: constants are improved, following a suggestion by B. Buk
A Dichotomy Theorem for Homomorphism Polynomials
In the present paper we show a dichotomy theorem for the complexity of
polynomial evaluation. We associate to each graph H a polynomial that encodes
all graphs of a fixed size homomorphic to H. We show that this family is
computable by arithmetic circuits in constant depth if H has a loop or no edge
and that it is hard otherwise (i.e., complete for VNP, the arithmetic class
related to #P). We also demonstrate the hardness over the rational field of cut
eliminator, a polynomial defined by B\"urgisser which is known to be neither VP
nor VNP-complete in the field of two elements, if VP is not equal to VNP (VP is
the class of polynomials computable by arithmetic circuit of polynomial size)
On Tractable Exponential Sums
We consider the problem of evaluating certain exponential sums. These sums
take the form ,
where each x_i is summed over a ring Z_N, and f(x_1,...,x_n) is a multivariate
polynomial with integer coefficients. We show that the sum can be evaluated in
polynomial time in n and log N when f is a quadratic polynomial. This is true
even when the factorization of N is unknown. Previously, this was known for a
prime modulus N. On the other hand, for very specific families of polynomials
of degree \ge 3, we show the problem is #P-hard, even for any fixed prime or
prime power modulus. This leads to a complexity dichotomy theorem - a complete
classification of each problem to be either computable in polynomial time or
#P-hard - for a class of exponential sums. These sums arise in the
classifications of graph homomorphisms and some other counting CSP type
problems, and these results lead to complexity dichotomy theorems. For the
polynomial-time algorithm, Gauss sums form the basic building blocks. For the
hardness results, we prove group-theoretic necessary conditions for
tractability. These tests imply that the problem is #P-hard for even very
restricted families of simple cubic polynomials over fixed modulus N
Quantum Discord and Quantum Computing - An Appraisal
We discuss models of computing that are beyond classical. The primary
motivation is to unearth the cause of nonclassical advantages in computation.
Completeness results from computational complexity theory lead to the
identification of very disparate problems, and offer a kaleidoscopic view into
the realm of quantum enhancements in computation. Emphasis is placed on the
`power of one qubit' model, and the boundary between quantum and classical
correlations as delineated by quantum discord. A recent result by Eastin on the
role of this boundary in the efficient classical simulation of quantum
computation is discussed. Perceived drawbacks in the interpretation of quantum
discord as a relevant certificate of quantum enhancements are addressed.Comment: To be published in the Special Issue of the International Journal of
Quantum Information on "Quantum Correlations: entanglement and beyond." 11
pages, 4 figure
An approximation trichotomy for Boolean #CSP
We give a trichotomy theorem for the complexity of approximately counting the
number of satisfying assignments of a Boolean CSP instance. Such problems are
parameterised by a constraint language specifying the relations that may be
used in constraints. If every relation in the constraint language is affine
then the number of satisfying assignments can be exactly counted in polynomial
time. Otherwise, if every relation in the constraint language is in the
co-clone IM_2 from Post's lattice, then the problem of counting satisfying
assignments is complete with respect to approximation-preserving reductions in
the complexity class #RH\Pi_1. This means that the problem of approximately
counting satisfying assignments of such a CSP instance is equivalent in
complexity to several other known counting problems, including the problem of
approximately counting the number of independent sets in a bipartite graph. For
every other fixed constraint language, the problem is complete for #P with
respect to approximation-preserving reductions, meaning that there is no fully
polynomial randomised approximation scheme for counting satisfying assignments
unless NP=RP
Path Coupling Using Stopping Times and Counting Independent Sets and Colourings in Hypergraphs
We give a new method for analysing the mixing time of a Markov chain using
path coupling with stopping times. We apply this approach to two hypergraph
problems. We show that the Glauber dynamics for independent sets in a
hypergraph mixes rapidly as long as the maximum degree Delta of a vertex and
the minimum size m of an edge satisfy m>= 2Delta+1. We also show that the
Glauber dynamics for proper q-colourings of a hypergraph mixes rapidly if m>= 4
and q > Delta, and if m=3 and q>=1.65Delta. We give related results on the
hardness of exact and approximate counting for both problems.Comment: Simpler proof of main theorem. Improved bound on mixing time. 19
page
The complexity of weighted boolean #CSP*
This paper gives a dichotomy theorem for the complexity of computing the partition
function of an instance of a weighted Boolean constraint satisfaction problem. The problem
is parameterized by a finite set F of nonnegative functions that may be used to assign weights to
the configurations (feasible solutions) of a problem instance. Classical constraint satisfaction problems
correspond to the special case of 0,1-valued functions. We show that computing the partition
function, i.e., the sum of the weights of all configurations, is FP#P-complete unless either (1) every
function in F is of “product type,” or (2) every function in F is “pure affine.” In the remaining cases,
computing the partition function is in P
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