For a weighted graph with n vertices and m edges, the Minimum k-Way Cut problem is to find a partition of the vertices into k sets that minimizes the total weight of edges crossing the sets. We obtain several important structural properties of minimum multiway cuts and use them to design efficient algorithms for several multiway partition problems. We design the first algorithm for finding minimum 3-way cuts in hypergraphs, which runs in O(dmn 3) time, where d is the sum of the degrees of all the vertices. We also give an O(n 4k--lg k) algorithm for finding all minimum k-way cuts in graphs. Our algorithm is based on a divide-and-conquer method and improves all well-known existing algorithms along this divide-and-conquer method. As for approximation algorithms, we determine the tight approximation ratio of a general greedy splitting algorithm (finding a minimum k-way cut by iteratively increasing a constant number of components). Our result implies that the approximation ratio of the algorithm that iteratively increases h -- 1 components is 2 -- h/k + O(h2 /k2), which settles a well-known open problem. / For an unweighted graph and a given subset T ⊂ V of k terminals, the Edge (respectively, Vertex) Multiterminal Cut problem is to find a set of l edges (respectively, non-terminal vertices), whose removal from G separates each terminal from all the others. We show that Edge Multiterminal Cut is polynomial-time solvable for 1 = O(log n) by presenting an O(2lkT(n, m)) algorithm, where T(n, m) is the running time of finding a maximum flow in unweighted graphs. We also give three algorithms for Vertex Multiterminal Cut that run in O(k lT(n, m)), O( l!2 2l T(n, m)) and O(lk 4lT( n, m)) time respectively. Furthermore, we obtain faster algorithms for small k: Edge 3-Terminal Cut can be solved in O(1.415lT(n, m)) time, and Vertex {3, 4, 5, 6}-Terminal Cuts can be solved in O(2.059 lT(n, m)), O(2.772 lT(n, m)), O(3.349 lT(n, m)) and O(3.857 lT(n, m)) times respectively. Our results on Multiterminal Cut can be used to obtain faster algorithms for Multicut. / In this thesis, we study algorithmic issues for three closely related partition problems in graphs: k-Way Cut (k-Cut), Multiterminal Cut, and Multicut. These three problems attempt to separate a graph, by edge or vertex deletion, into several components with certain properties. The k-Way Cut (k-Cut) problem is to separate the graph into k components, the Multiterminal Cut problem is to separate a subset of vertices away from each other, and the Multicut problem is to separate some given pairs of vertices. These three problems have many applications in parallel and distributed computing, VLSI system design, clustering problems, communications network and many others. / Xiao, Mingyu. / Adviser: Andrew C. Yao. / Source: Dissertation Abstracts International, Volume: 70-06, Section: B, page: 3617. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 85-92). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_344306 |
| Date | January 2008 |
| Contributors | Xiao, Mingyu., Chinese University of Hong Kong Graduate School. Division of Computer Science and Engineering. |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, theses |
| Format | electronic resource, microform, microfiche, 1 online resource (viii, 92 leaves : ill.) |
| Rights | Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/) |
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