38 research outputs found
A Simple FPTAS for Counting Edge Covers
An edge cover of a graph is a set of edges such that every vertex has at
least an adjacent edge in it. Previously, approximation algorithm for counting
edge covers is only known for 3 regular graphs and it is randomized. We design
a very simple deterministic fully polynomial-time approximation scheme (FPTAS)
for counting the number of edge covers for any graph. Our main technique is
correlation decay, which is a powerful tool to design FPTAS for counting
problems. In order to get FPTAS for general graphs without degree bound, we
make use of a stronger notion called computationally efficient correlation
decay, which is introduced in [Li, Lu, Yin SODA 2012].Comment: To appear in SODA 201
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
The Complexity of Contracting Planar Tensor Network
Tensor networks have been an important concept and technique in many research
areas such as quantum computation and machine learning. We study the complexity
of evaluating the value of a tensor network. This is also called contracting
the tensor network. In this article, we focus on computing the value of a
planar tensor network where every tensor specified at a vertex is a Boolean
symmetric function. We design two planar gadgets to obtain a sub-exponential
time algorithm. The key is to remove high degree vertices while essentially not
changing the size of the tensor network. The algorithm runs in time
. Furthermore, we use a counting version of the
Sparsification Lemma to prove a matching lower bound
assuming \#ETH holds
On the Complexity of #CSP^d
Counting CSP^d is the counting constraint satisfaction problem (#CSP in short) restricted to the instances where every variable occurs a multiple of d times. This paper revisits tractable structures in #CSP and gives a complexity classification theorem for #CSP^d with algebraic complex weights. The result unifies affine functions (stabilizer states in quantum information theory) and related variants such as the local affine functions, the discovery of which leads to all the recent progress on the complexity of Holant problems.
The Holant is a framework that generalizes counting CSP. In the literature on Holant problems, weighted constraints are often expressed as tensors (vectors) such that projections and linear transformations help analyze the structure. This paper gives an example showing that different classes of tensors distinguished by these algebraic operations may share the same closure property under tensor product and contraction
The #CSP Dichotomy is Decidable
Bulatov (2008) and Dyer and Richerby (2010) have established the following dichotomy for the counting constraint satisfaction problem (#CSP): for any constraint language Gamma, the
problem of computing the number of satisfying assignments to constraints drawn from Gamma is either in FP or is #P-complete, depending on the structure of Gamma. The principal question left open by this research was whether the criterion of the dichotomy is decidable. We show that it is; in fact, it is in NP
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