| Summary: | 博士 === 國立清華大學 === 資訊科學研究所 === 78 === A Bottleneck optimization problem on general graphs with edge costs is the
problem of finding a subgraph of a certain kind that minimizes the mzximum
edge cost in the subgraph. a Euclidean bottleneck optimization problem is
a bottleneck optimization problem on complete graphs which are constructed
from a set of points in the plane and whose edge cost are Euclidean
distances between points connected by edges. In this dissertation, we
define a special graph called k7 relative Neighborhood Graph, denoted as
kRNG, where k is a positive number, and use it to solve the following
three Euclidean bottleneck optimization problems:
(A) The Euclidean bottleneck matching problem.
(B) The Euclidean bottleneck biconnected edge subgraph problem.
(C) The Euclidean bottleneck traveling salesperson problem.
We prove the following three theorems:
(1) For any instance of Problem A, there exists an optimal solution which
is a subgraph of a 17RNG.
(2) For any instance of Problem A, there exists an optimal solution which
is a subgraph of a 2RNG.
(3) For any instance of Problem A, there exists an optimal solution which
is a subgraph of a 20RNG.
All numbers of edges of these three special graphs are O(n). Therefore we
can find optimal solutions for the above three problems from three
k7 relative neighborhood graphs. In this way, we can solve Problem A and
Problem B in O(n2)time, and also an efficient approximation algorithm for
Problem C is developed. The third theorem above gives us an interesting
graph theoretic result: 20RNGs are Hamiltonian.
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