arXiv subjects

Kuan Yang

Publications and source records attributed to Kuan Yang.

21 records · Page 2Linked to original sources

arXiv2016

An FPTAS for Counting Proper Four-Colorings on Cubic Graphs

Graph coloring is arguably the most exhaustively studied problem in the area of approximate counting. It is conjectured that there is a fully polynomial-time (randomized) approximation scheme (FPTAS/FPRAS) for counting the number of proper colorings as long as $q \geq Δ+ 1$, where $q$ is the number of colors and $Δ$ is the maximum degree of the graph. The bound of $q = Δ+ 1$ is the uniqueness threshold for Gibbs measure on $Δ$-regular infinite trees. However, the conjecture remained open even for any fixed $Δ\geq 3$ (The cases of $Δ=1, 2$ are trivial). In this paper, we design an FPTAS for counting the number of proper $4$-colorings on graphs with maximum degree $3$ and thus confirm the conjecture in the case of $Δ=3$. This is the first time to achieve this optimal bound of $q = Δ+ 1$. Previously, the best FPRAS requires $q > \frac{11}{6} Δ$ and the best deterministic FPTAS requires $q > 2.581Δ+ 1$ for general graphs. In the case of $Δ=3$, the best previous result is an FPRAS for counting proper 5-colorings. We note that there is a barrier to go beyond $q = Δ+ 2$ for single-site Glauber dynamics based FPRAS and we overcome this by correlation decay approach. Moreover, we develop a number of new techniques for the correlation decay approach which can find applications in other approximate counting problems.

cs.DS↗

arXiv2015

FPTAS for Hardcore and Ising Models on Hypergraphs

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.

cs.DS↗

arXiv2014

Graph metric with no proper inclusion between lines

In trying to generalize the classic Sylvester-Gallai theorem and De Bruijn-Erdős theorem in plane geometry, lines and closure lines were previously defined for metric spaces and hypergraphs. Both definitions do not obey the geometric intuition in the sense that two lines (closure lines) may intersect at more than one point, and one line (closure line) might be the proper subset of another. In this work, we study the systems where one or both of the configurations are forbidden. We note that when any two lines intersect in at most one point, the two classic theorems extend in any metric space. We study the metric spaces induced by simple graphs where no line is a proper subset of another, and show that the least number of lines for such a graph with $n$ vertices is between the order of $n^{4/3}$ and $n^{4/3} \ln^{2/3} n$.

math.MG↗