We prove a complexity classification theorem that classifies all counting constraint satisfaction problems (#CSP) over Boolean variables into exactly three categories: (1) Polynomial-time tractable; (2) #P-hard for general instances, but solvable in polynomial-time over planar graphs; and (3) #P-hard over planar graphs. The classification applies to all… (More)
Holographic algorithms with matchgates are a novel approach to design polynomial time computation. It uses Kasteleyn's algorithm for perfect matchings, and more importantly a holographic reduction. The two fundamental parameters of a holographic reduction are the domain size k of the underlying problem, and the basis size ℓ. A holographic reduction… (More)
We prove a complexity dichotomy for complex-weighted Holant problems with an arbitrary set of symmetric constraint functions on Boolean variables. In the study of counting complexity, such as #CSP, there are problems which are #P-hard over general graphs but P-time solvable over planar graphs. A recurring theme has been that a holographic reduction  to… (More)
We prove a complexity dichotomy theorem for the six-vertex model. For every setting of the parameters of the model, we prove that computing the partition function is either solvable in polynomial time or #P-hard. The dichotomy criterion is explicit.
We prove a complexity classification theorem that divides the six-vertex model into exactly three types. For every setting of the parameters of the model, the computation of the partition function is precisely: (1) Solvable in polynomial time for every graph, or (2) #P-hard for general graphs but solvable in polynomial time for planar graphs, or (3) #P-hard… (More)
We prove a complexity dichotomy theorem for the eight-vertex model. For every setting of the parameters of the model, we prove that computing the partition function is either solvable in polynomial time or #P-hard. The dichotomy criterion is explicit. For tractability, we find some new classes of problems computable in polynomial time. For #P-hardness, we… (More)
BACKGROUND This study aimed to compare the surgical outcomes between open pedicle screw fixation (OPSF) and percutaneous pedicle screw fixation (PPSF) for the treatment of thoracolumbar fractures, which has received scant research attention to date. MATERIAL AND METHODS Eight-four patients with acute and subacute thoracolumbar fractures who were treated… (More)
Research Interests • Computational complexity. Exact and approximate counting algorithms and hardness.
An essential problem in the design of holographic algorithms is to decide whether the required signatures can be realized by matchgates under a suitable basis transformation (SRP). For holographic algorithms on domain size 2, [1, 2, 4, 5] have built a systematical theory. In this paper, we reduce SRP on domain size k ≥ 3 to SRP on domain size 2 for… (More)