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Algorithms for flows and disjoint paths in planar graphs
Wang, Yipu
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https://hdl.handle.net/2142/108483
Description
- Title
- Algorithms for flows and disjoint paths in planar graphs
- Author(s)
- Wang, Yipu
- Issue Date
- 2020-07-14
- Director of Research (if dissertation) or Advisor (if thesis)
- Erickson, Jeff
- Doctoral Committee Chair(s)
- Erickson, Jeff
- Committee Member(s)
- Chekuri, Chandra
- Chandrasekaran, Karthekeyan
- Klein, Philip N
- Department of Study
- Computer Science
- Discipline
- Computer Science
- Degree Granting Institution
- University of Illinois at Urbana-Champaign
- Degree Name
- Ph.D.
- Degree Level
- Dissertation
- Keyword(s)
- planar graphs
- maximum flow
- element connectivity
- scaling algorithm
- disjoint paths
- orientation
- ideal orientation
- approximation algorithms
- reduced graph
- Abstract
- In this dissertation we describe several algorithms for computing flows, connectivity, and disjoint paths in planar graphs. In all cases, the algorithms are either the first polynomial-time algorithms or are faster than all previously-known algorithms. First, we describe algorithms for the maximum flow problem in directed planar graphs with integer capacities on both vertices and arcs and with multiple sources and sinks. The algorithms are the first to solve the problem in near-linear time when the number of terminals is fixed and the capacities are polynomially bounded. As a byproduct, we get the first algorithm to solve the vertex-disjoint S-T paths problem in near-linear time when the number of terminals is fixed but greater than 2. We also modify our algorithms to handle real capacities in near-linear time when they are three terminals. Second, we describe algorithms to compute element-connectivity and a related structure called the reduced graph. We show that global element-connectivity in planar graphs can be found in linear time if the terminals can be covered by O(1) faces. We also show that the reduced graph can be computed in subquadratic time in planar graphs if the number of terminals is fixed. Third, we describe algorithms for solving or approximately solving the vertex-disjoint paths problem when we want to minimize the total length of the paths. For planar graphs, we describe: (1) an exact algorithm for the case of four pairs of terminals on a single face; and (2) a k-approximation algorithm for the case of k pairs of terminals on a single face. Fourth, we describe algorithms and a hardness result for the ideal orientation problem. We show that the problem is NP-hard in planar graphs. On the other hand, we show that the problem is polynomial-time solvable in planar graphs when the number of terminals is fixed, the terminals are all on the same face, and no two of the terminal pairs cross. We also describe an algorithm for serial instances of a generalization of the ideal orientation problem called the k-min-sum orientation problem.
- Graduation Semester
- 2020-08
- Type of Resource
- Thesis
- Permalink
- http://hdl.handle.net/2142/108483
- Copyright and License Information
- Copyright 2020 Yipu Wang
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Graduate Dissertations and Theses at Illinois PRIMARY
Graduate Theses and Dissertations at IllinoisDissertations and Theses - Computer Science
Dissertations and Theses from the Dept. of Computer ScienceManage Files
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