Glauber Dynamics for Sampling an Edge Colouring of Full Homogeneous Trees

• Other Authors: Poon, Chun Yeung (author.)
• Format: Others
• Language: English; Chinese
• Published: 2016
• Online Access: http://repository.lib.cuhk.edu.hk/en/item/cuhk-1292280

我們研究用 Glauber Dynamics 隨機抽出圖頂點著色的問題。這是個有趣的問題因為 Jerrum 證明如果能隨機抽出頂點著色，則可以找出圖形著色數目的近似值。我們希望有 多項式時間內的取樣器。Glauber Dynamics 用馬可夫鏈抽出圖形著色並被大量研究。它 的狀態空間是頂點著色的集合。每一步我們隨機抽出一種顏色 c 和頂點 u，並嘗試把 u 的 顏色改成 c。如果有衝突則停留在現在的顏色。假設圖的最大度是 d，顏色數目是 q。這 問題其中一個目標是証明當 q 不少於 d + 2 時，混合時間是多項式時間。現在最好的成果 是 Vigoda 提出的對於所有圖，混合時間是多項式時間如果 q 不少於 11d/6。對於某一種 類的圖，我們可以用更少的顏色。其中一個例子是有大周長和大的最大度的圖。 === 在這篇論文，我們集中在用 Glauber Dynamics 隨機抽出正則樹邊著色的問題。這相當於 用 Glauber Dynamics 隨機抽出正則樹的線圖的頂點著色。我們研究這特殊例子因為線圖 的周長小因此不能直接應用之前的結果。目前最好的結果是 vigoda 的多項式混合時間如 果 q 不少於 11d/3。我們的主要成果是多項式混合時間如果 q 不少於 2d。 === We study the problem of sampling a graph colouring using Glauber Dynamics. This is an interesting problem as Jerrum showed that we can approximate the number of proper colouring if we can sample a colouring nearly uniformly. Therefore we want a sampler with polynomial running time. Glauber Dynamics is one natural Markov Chain for sampling a graph colouring and it has been studied extensively. The state space of it is the set of proper colourings. In each step, we sample a colour c and a node u randomly. Then we update u to colour c if the new colouring is still proper, otherwise we stay at the current colouring. For a graph with maximum degree d, let q be the number of colours. One important goal on this problem is proving polynomial mixing time if q ≥ d + 2. For general graphs, the best result is polynomial mixing time if q ≥ 11d/6 by Vigoda. For some classes of graphs, we can sample a colouring for fewer number of colours. One example are graphs with large girth and large maximum degree and there are many results for this kinds of graphs. === In the thesis, we focus on the mixing time of Glauber Dynamics for sampling an edge colouring of a full d-homogeneous tree. This is equivalent to sampling a proper vertex colouring of the line graph of the tree. We consider this special case as the line graph has small girth so previous results and techniques does not apply directly. The best previous result is polynomial mixing time if q ≥ 11d/3 by Vigoda. Our main result is polynomial mixing time if q ≥ 2d. Our proof is based on the multicommodity flow argument by Sinclair. === Poon, Chun Yeung. === Thesis M.Phil. Chinese University of Hong Kong 2016. === Includes bibliographical references (leaves ). === Abstracts also in Chinese. === Title from PDF title page (viewed on …). === Detailed summary in vernacular field only. === Detailed summary in vernacular field only.