The NP-Hardness of Covering Points with Lines, Paths and Tours and their Tractability with FPT-Algorithms

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Author(s)
Primary Supervisor
Estivill-Castro, Vladimir
Suraweera, Francis
Year published
2010
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Show full item recordAbstract
Given a problem for which no polynomial-time algorithm is likely to exist, we
investigate how to attack this seemingly intractable problem based on parameterized
complexity theory. We study hard geometric problems, and show that
they are xed-parameter tractable (FPT) given an instance and a parameter
k. This allows the problems to be solved exactly, rather than approximately, in
polynomial time in the size of the input and exponential time in the parameter.
Although the parameterized approach is still young, in recent years, there
have been many results published concerning graph problems and databases.
However, not many ...
View more >Given a problem for which no polynomial-time algorithm is likely to exist, we investigate how to attack this seemingly intractable problem based on parameterized complexity theory. We study hard geometric problems, and show that they are xed-parameter tractable (FPT) given an instance and a parameter k. This allows the problems to be solved exactly, rather than approximately, in polynomial time in the size of the input and exponential time in the parameter. Although the parameterized approach is still young, in recent years, there have been many results published concerning graph problems and databases. However, not many earlier results apply the parameterized approach in the eld of computational geometry. This thesis, therefore, focuses on geometric NP-hard problems. These problems are the Line Cover problem, the Rectilinear Line Cover problem in higher dimensions, the Rectilinear Minimum-Links Spanning Path problem in higher dimensions, the Rectilinear Hyper- plane Cover problem, the Minimum-Bends Traveling Salesman Prob- lem and the Rectilinear Minimum-Bends Traveling Salesman Prob- lem, in addition to some other variants of these problems. The Rectilinear Minimum-Links Spanning Path problem in higher dimensions and the Rectilinear Hyperplane Cover problem had been the subject of only conjectures about their intractability. Therefore, we present the NP-completeness proofs for these problems. After verifying their hardness, we use the xed-parameter approach to solve the two problems. We focus on solving the decision version of the problems, rather than solving the optimizations. However, with the Line Cover problem we demonstrate that it is not dicult to adapt algorithms for the decision version to algorithms for the optimization version. We also implement several algorithms for the Line Cover problem and conduct experimental evaluations of our algorithms with respect to previously known algorithms. For the remaining problems in the thesis, we will establish only the fundamental results. That is, we determine xed-parameter tractability of those problems.
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View more >Given a problem for which no polynomial-time algorithm is likely to exist, we investigate how to attack this seemingly intractable problem based on parameterized complexity theory. We study hard geometric problems, and show that they are xed-parameter tractable (FPT) given an instance and a parameter k. This allows the problems to be solved exactly, rather than approximately, in polynomial time in the size of the input and exponential time in the parameter. Although the parameterized approach is still young, in recent years, there have been many results published concerning graph problems and databases. However, not many earlier results apply the parameterized approach in the eld of computational geometry. This thesis, therefore, focuses on geometric NP-hard problems. These problems are the Line Cover problem, the Rectilinear Line Cover problem in higher dimensions, the Rectilinear Minimum-Links Spanning Path problem in higher dimensions, the Rectilinear Hyper- plane Cover problem, the Minimum-Bends Traveling Salesman Prob- lem and the Rectilinear Minimum-Bends Traveling Salesman Prob- lem, in addition to some other variants of these problems. The Rectilinear Minimum-Links Spanning Path problem in higher dimensions and the Rectilinear Hyperplane Cover problem had been the subject of only conjectures about their intractability. Therefore, we present the NP-completeness proofs for these problems. After verifying their hardness, we use the xed-parameter approach to solve the two problems. We focus on solving the decision version of the problems, rather than solving the optimizations. However, with the Line Cover problem we demonstrate that it is not dicult to adapt algorithms for the decision version to algorithms for the optimization version. We also implement several algorithms for the Line Cover problem and conduct experimental evaluations of our algorithms with respect to previously known algorithms. For the remaining problems in the thesis, we will establish only the fundamental results. That is, we determine xed-parameter tractability of those problems.
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Thesis Type
Thesis (PhD Doctorate)
Degree Program
Doctor of Philosophy (PhD)
School
School of Information and Communication Technology
Copyright Statement
The author owns the copyright in this thesis, unless stated otherwise.
Item Access Status
Public
Subject
algorithm
polynomial-time
line cover
rectilinear