8,842 research outputs found
A complexity dichotomy for partition functions with mixed signs
Partition functions, also known as homomorphism functions, form a rich family
of graph invariants that contain combinatorial invariants such as the number of
k-colourings or the number of independent sets of a graph and also the
partition functions of certain "spin glass" models of statistical physics such
as the Ising model.
Building on earlier work by Dyer, Greenhill and Bulatov, Grohe, we completely
classify the computational complexity of partition functions. Our main result
is a dichotomy theorem stating that every partition function is either
computable in polynomial time or #P-complete. Partition functions are described
by symmetric matrices with real entries, and we prove that it is decidable in
polynomial time in terms of the matrix whether a given partition function is in
polynomial time or #P-complete.
While in general it is very complicated to give an explicit algebraic or
combinatorial description of the tractable cases, for partition functions
described by a Hadamard matrices -- these turn out to be central in our proofs
-- we obtain a simple algebraic tractability criterion, which says that the
tractable cases are those "representable" by a quadratic polynomial over the
field GF(2)
The Complexity of Weighted Boolean #CSP with Mixed Signs
We give a complexity dichotomy for the problem of computing the partition
function of a weighted Boolean constraint satisfaction problem. Such a problem
is parameterized by a set of rational-valued functions, which generalize
constraints. Each function assigns a weight to every assignment to a set of
Boolean variables. Our dichotomy extends previous work in which the weight
functions were restricted to being non-negative. We represent a weight function
as a product of the form (-1)^s g, where the polynomial s determines the sign
of the weight and the non-negative function g determines its magnitude. We show
that the problem of computing the partition function (the sum of the weights of
all possible variable assignments) is in polynomial time if either every weight
function can be defined by a "pure affine" magnitude with a quadratic sign
polynomial or every function can be defined by a magnitude of "product type"
with a linear sign polynomial. In all other cases, computing the partition
function is FP^#P-complete.Comment: 24 page
The complexity of weighted and unweighted #CSP
We give some reductions among problems in (nonnegative) weighted #CSP which
restrict the class of functions that needs to be considered in computational
complexity studies. Our reductions can be applied to both exact and approximate
computation. In particular, we show that a recent dichotomy for unweighted #CSP
can be extended to rational-weighted #CSP.Comment: 11 page
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
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