24 research outputs found
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
Clifford Gates in the Holant Framework
We show that the Clifford gates and stabilizer circuits in the quantum
computing literature, which admit efficient classical simulation, are
equivalent to affine signatures under a unitary condition. The latter is a
known class of tractable functions under the Holant framework.Comment: 14 page
The Complexity of Holant Problems over Boolean Domain with Non-Negative Weights
Holant problem is a general framework to study the computational complexity of counting problems. We prove a complexity dichotomy theorem for Holant problems over the Boolean domain with non-negative weights. It is the first complete Holant dichotomy where constraint functions are not necessarily symmetric.
Holant problems are indeed read-twice #CSPs. Intuitively, some #CSPs that are #P-hard become tractable when restricted to read-twice instances. To capture them, we introduce the Block-rank-one condition. It turns out that the condition leads to a clear separation. If a function set F satisfies the condition, then F is of affine type or product type. Otherwise (a) Holant(F) is #P-hard; or (b) every function in F is a tensor product of functions of arity at most 2; or (c) F is transformable to a product type by some real orthogonal matrix. Holographic transformations play an important role in both the hardness proof and the characterization of tractability
A complete dichotomy for complex-valued holant<sup>c</sup>
Holant problems are a family of counting problems on graphs, parametrised by sets of complex-valued functions of Boolean inputs. Holant^c denotes a subfamily of those problems, where any function set considered must contain the two unary functions pinning inputs to values 0 or 1. The complexity classification of Holant problems usually takes the form of dichotomy theorems, showing that for any set of functions in the family, the problem is either #P-hard or it can be solved in polynomial time. Previous such results include a dichotomy for real-valued Holant^c and one for Holant^c with complex symmetric functions, i.e. functions which only depend on the Hamming weight of the input. Here, we derive a dichotomy theorem for Holant^c with complex-valued, not necessarily symmetric functions. The tractable cases are the complex-valued generalisations of the tractable cases of the real-valued Holant^c dichotomy. The proof uses results from quantum information theory, particularly about entanglement. This full dichotomy for Holant^c answers a question that has been open for almost a decade.</p