Abstract:
We propose a notion of contraction function for a family of graphs and establish its connection to the strong spatial mixing for spin systems. More specifically, we show that for anti-ferromagnetic Potts model on families of graphs characterized by a specific contraction function, the model exhibits strong spatial mixing, and if further the graphs exhibit certain local sparsity which are very natural and easy to satisfy by typical sparse graphs, then we also have FPTAS for computing the partition function. This new characterization of strong spatial mixing of multi-spin system does not require maximum degree of the graphs to be bounded, but instead it relates the decay of correlation of the model to a notion of effective average degree measured by the contraction of a function on the family of graphs. It also generalizes other notion of effective average degree which may determine the strong spatial mixing, such as the connective constant, whose connection to strong spatial mixing is only known for very simple models and is not extendable to general spin systems. As direct consequences: (1) we obtain FPTAS for the partition function of $q$-state anti-ferromagnetic Potts model with activity $0\le\beta<1$ on graphs of maximum degree bounded by $d$ when $q> 3(1-\beta)d+1$, improving the previous best bound $\beta> 3(1-\beta)d$ and asymptotically approaching the inapproximability threshold $q=(1-\beta)d$, and (2) we obtain an efficient sampler (in the same sense of fully polynomial-time almost uniform sampler, FPAUS) for the Potts model on Erd\H{o}s-R\'enyi random graph $\mathcal{G}(n,d/n)$ with sufficiently large constant $d$, provided that $q> 3(1-\beta)d+4$. In particular when $\beta=0$, the sampler becomes an FPAUS for for proper $q$-coloring in $\mathcal{G}(n,d/n)$ with $q> 3d+4$, improving the current best bound $q> 5.5d$ for FPAUS for $q$-coloring in $\mathcal{G}(n,d/n)$.

Abstract:
We show for a broad class of counting problems, correlation decay (strong spatial mixing) implies FPTAS on planar graphs. The framework for the counting problems considered by us is the Holant problems with arbitrary constant-size domain and symmetric constraint functions. We define a notion of regularity on the constraint functions, which covers a wide range of natural and important counting problems, including all multi-state spin systems, counting graph homomorphisms, counting weighted matchings or perfect matchings, the subgraphs world problem transformed from the ferromagnetic Ising model, and all counting CSPs and Holant problems with symmetric constraint functions of constant arity. The core of our algorithm is a fixed-parameter tractable algorithm which computes the exact values of the Holant problems with regular constraint functions on graphs of bounded treewidth. By utilizing the locally tree-like property of apex-minor-free families of graphs, the parameterized exact algorithm implies an FPTAS for the Holant problem on these graph families whenever the Gibbs measure defined by the problem exhibits strong spatial mixing. We further extend the recursive coupling technique to Holant problems and establish strong spatial mixing for the ferromagnetic Potts model and the subgraphs world problem. As consequences, we have new deterministic approximation algorithms on planar graphs and all apex-minor-free graphs for several counting problems.

Abstract:
The problem of sampling proper $q$-colorings from uniform distribution has been extensively studied. Most of existing samplers require $q\ge \alpha \Delta+\beta$ for some constants $\alpha$ and $\beta$, where $\Delta$ is the maximum degree of the graph. The problem becomes more challenging when the underlying graph has unbounded degree since even the decision of $q$-colorability becomes nontrivial in this situation. The Erd\H{o}s-R\'{e}nyi random graph $\mathcal{G}(n,d/n)$ is a typical class of such graphs and has received a lot of recent attention. In this case, the performance of a sampler is usually measured by the relation between $q$ and the average degree $d$. We are interested in the fully polynomial-time almost uniform sampler (FPAUS) and the state-of-the-art with such sampler for proper $q$-coloring on $\mathcal{G}(n,d/n)$ requires that $q\ge 5.5d$. In this paper, we design an FPAUS for proper $q$-colorings on $\mathcal{G}(n,d/n)$ by requiring that $q\ge 3d+O(1)$, which improves the best bound for the problem so far. Our sampler is based on the spatial mixing property of $q$-coloring on random graphs. The core of the sampler is a deterministic algorithm to estimate the marginal probability on blocks, which is computed by a novel block version of recursion for $q$-coloring on unbounded degree graphs.

Abstract:
We study the approximability of computing the partition function for ferromagnetic two-state spin systems. The remarkable algorithm by Jerrum and Sinclair showed that there is a fully polynomial-time randomized approximation scheme (FPRAS) for the special ferromagnetic Ising model with any given uniform external field. Later, Goldberg and Jerrum proved that it is #BIS-hard for Ising model if we allow inconsistent external fields on different nodes. In contrast to these two results, we prove that for any ferromagnetic two-state spin systems except the Ising model, there exists a threshold for external fields beyond which the problem is #BIS-hard, even if the external field is uniform.

Abstract:
Fibonacci gate problems have severed as computation primitives to solve other problems by holographic algorithm and play an important role in the dichotomy of exact counting for Holant and CSP frameworks. We generalize them to weighted cases and allow each vertex function to have different parameters, which is a much boarder family and #P-hard for exactly counting. We design a fully polynomial-time approximation scheme (FPTAS) for this generalization by correlation decay technique. This is the first deterministic FPTAS for approximate counting in the general Holant framework without a degree bound. We also formally introduce holographic reduction in the study of approximate counting and these weighted Fibonacci gate problems serve as computation primitives for approximate counting. Under holographic reduction, we obtain FPTAS for other Holant problems and spin problems. One important application is developing an FPTAS for a large range of ferromagnetic two-state spin systems. This is the first deterministic FPTAS in the ferromagnetic range for two-state spin systems without a degree bound. Besides these algorithms, we also develop several new tools and techniques to establish the correlation decay property, which are applicable in other problems.

Abstract:
Hardcore and Ising models are two most important families of two state spin systems in statistic physics. Partition function of spin systems is the center concept in statistic physics which connects microscopic particles and their interactions with their macroscopic and statistical properties of materials such as energy, entropy, ferromagnetism, etc. If each local interaction of the system involves only two particles, the system can be described by a graph. In this case, fully polynomial-time approximation scheme (FPTAS) for computing the partition function of both hardcore and anti-ferromagnetic Ising model was designed up to the uniqueness condition of the system. These result are the best possible since approximately computing the partition function beyond this threshold is NP-hard. In this paper, we generalize these results to general physics systems, where each local interaction may involves multiple particles. Such systems are described by hypergraphs. For hardcore model, we also provide FPTAS up to the uniqueness condition, and for anti-ferromagnetic Ising model, we obtain FPTAS where a slightly stronger condition holds.

Abstract:
Markov Chain Monte Carlo (MCMC) method is a widely used algorithm design scheme with many applications. To make efficient use of this method, the key step is to prove that the Markov chain is rapid mixing. Canonical paths is one of the two main tools to prove rapid mixing. However, there are much fewer success examples comparing to coupling, the other main tool. The main reason is that there is no systematic approach or general recipe to design canonical paths. Building up on a previous exploration by McQuillan, we develop a general theory to design canonical paths for MCMC: We reduce the task of designing canonical paths to solving a set of linear equations, which can be automatically done even by a machine. Making use of this general approach, we obtain fully polynomial-time randomized approximation schemes (FPRAS) for counting the number of $b$-matching with $b\leq 7$ and $b$-edge-cover with $b\leq 2$. They are natural generalizations of matchings and edge covers for graphs. No polynomial time approximation was previously known for these problems.

Abstract:
Structural and functional information encoded in DNA combined with unique properties of nanomaterials could be of use for the construction of novel biocomputational circuits and intelligent biomedical nanodevices. However, at present their practical applications are still limited by either low reproducibility of fabrication, modest sensitivity, or complicated handling procedures. Here, we demonstrate the construction of label-free and switchable molecular logic gates that use specific conformation modulation of a guanine- and thymine- rich DNA, while the optical readout is enabled by the tunable alphabetical metamaterials, which serve as a substrate for surface enhanced Raman spectroscopy (MetaSERS). By computational and experimental investigations, we present a comprehensive solution to tailor the plasmonic responses of MetaSERS with respect to the metamaterial geometry, excitation energy, and polarization. Our tunable MetaSERS-based DNA logic is simple to operate, highly reproducible, and can be stimulated by ultra-low concentration of the external inputs, enabling an extremely sensitive detection of mercury ions.

Abstract:
This paper presents a pivoting-based method for solving convex quadratic programming and then shows how to use it together with a parameter technique to solve mean-variance portfolio selection problems.

Abstract:
At present, sexual harassment in domestic workplace has a high probability of occurrence, which causes more and more attention. In this paper, the form of sexual harassment in workplace, and how to solve the sexual harassment were investigated and analyzed through questionnaires; and countermeasures and management suggestions were put forward from three aspects of corporate, employees and family.