A finite dominating set of cardinality O(k) and a witness set of cardinality O(n) for 1.5D terrain guarding problem
1.5 dimensional (1.5D) terrain is characterized by a piecewise linear curve. Locating minimum number of guards on the terrain ( T ) to cover/guard the whole terrain is known as 1.5D terrain guarding problem. Approximation algorithms and a polynomial-time approximation scheme have been presented for...
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| Vydáno v: | Annals of operations research Ročník 254; číslo 1-2; s. 37 - 46 |
|---|---|
| Hlavní autor: | |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
Springer US
01.07.2017
Springer Springer Nature B.V |
| Témata: | |
| ISSN: | 0254-5330, 1572-9338 |
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| Abstract | 1.5 dimensional (1.5D) terrain is characterized by a piecewise linear curve. Locating minimum number of guards on the terrain (
T
) to cover/guard the whole terrain is known as 1.5D terrain guarding problem. Approximation algorithms and a polynomial-time approximation scheme have been presented for the problem. The problem has been shown to be NP-Hard. In the problem, the set of possible guard locations and the set of points to be guarded are uncountable. To solve the problem to optimality, a finite dominating set (FDS) of size
O
(
n
2
)
and a witness set of size
O
(
n
3
)
have been presented, where
n
is the number of vertices on
T
. We show that there exists an even smaller FDS of cardinality
O
(
k
)
and a witness set of cardinality O(
n
), where
k
is the number of convex points. Convex points are vertices with the additional property that between any two convex points the piecewise linear curve representing the terrain is convex. Since it is always true that
k
≤
n
for
n
≥
2 and since it is possible to construct terrains such that
n
=
2
k
, the existence of an FDS with cardinality O(
k
) and a witness set of cardinality of
O
(
n
)
leads to the reduction of decision variables and constraints respectively in the zero-one integer programming formulation of the problem. |
|---|---|
| AbstractList | 1.5 dimensional (1.5D) terrain is characterized by a piecewise linear curve. Locating minimum number of guards on the terrain (
T
) to cover/guard the whole terrain is known as 1.5D terrain guarding problem. Approximation algorithms and a polynomial-time approximation scheme have been presented for the problem. The problem has been shown to be NP-Hard. In the problem, the set of possible guard locations and the set of points to be guarded are uncountable. To solve the problem to optimality, a finite dominating set (FDS) of size
O
(
n
2
)
and a witness set of size
O
(
n
3
)
have been presented, where
n
is the number of vertices on
T
. We show that there exists an even smaller FDS of cardinality
O
(
k
)
and a witness set of cardinality O(
n
), where
k
is the number of convex points. Convex points are vertices with the additional property that between any two convex points the piecewise linear curve representing the terrain is convex. Since it is always true that
k
≤
n
for
n
≥
2 and since it is possible to construct terrains such that
n
=
2
k
, the existence of an FDS with cardinality O(
k
) and a witness set of cardinality of
O
(
n
)
leads to the reduction of decision variables and constraints respectively in the zero-one integer programming formulation of the problem. (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) 1.5 dimensional (1.5D) terrain is characterized by a piecewise linear curve. Locating minimum number of guards on the terrain (T) to cover/guard the whole terrain is known as 1.5D terrain guarding problem. Approximation algorithms and a polynomial-time approximation scheme have been presented for the problem. The problem has been shown to be NP-Hard. In the problem, the set of possible guard locations and the set of points to be guarded are uncountable. To solve the problem to optimality, a finite dominating set (FDS) of size ... and a witness set of size ... have been presented, where n is the number of vertices on T. We show that there exists an even smaller FDS of cardinality ... and a witness set of cardinality O(n), where k is the number of convex points. Convex points are vertices with the additional property that between any two convex points the piecewise linear curve representing the terrain is convex. Since it is always true that ... for ... 2 and since it is possible to construct terrains such that ..., the existence of an FDS with cardinality O(k) and a witness set of cardinality of ... leads to the reduction of decision variables and constraints respectively in the zero-one integer programming formulation of the problem. |
| Audience | Academic |
| Author | Eliş, Haluk |
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| Cites_doi | 10.1023/A:1008395305189 10.1023/A:1018951502447 10.1287/opre.39.1.100 10.1080/00207160.2012.707800 10.1137/S0097539704446384 10.1007/s00454-006-1273-8 10.1016/0925-7721(95)00034-8 10.1016/S0020-0190(01)00255-1 10.1068/b12979 10.1007/BF02097802 10.1007/s10479-005-2048-y 10.1016/S0747-7171(89)80003-3 10.1137/100791506 10.1080/02693799408902004 10.1007/s00453-009-9358-4 10.1007/978-3-642-03685-9_11 10.1007/11682462_58 |
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10.1007/s10479-005-2048-y – ident: 2432_CR4 – volume: 37 start-page: 43 year: 2007 ident: 2432_CR5 publication-title: Discrete Computational Geometry doi: 10.1007/s00454-006-1273-8 – volume: 39 start-page: 100 issue: 1 year: 1991 ident: 2432_CR17 publication-title: Operations Research doi: 10.1287/opre.39.1.100 – volume: 7 start-page: 11 year: 1989 ident: 2432_CR6 publication-title: Journal of Symbolic Computation doi: 10.1016/S0747-7171(89)80003-3 – volume: 30 start-page: 709 issue: 5 year: 2003 ident: 2432_CR7 publication-title: Environment and Planning B: Planning and Design doi: 10.1068/b12979 |
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T
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| SubjectTerms | Algorithms Approximation Business and Management Combinatorics Convex functions Estimating techniques Guards Integer programming Location analysis Mathematical analysis Mathematical research Operations research Operations Research/Decision Theory Original Paper Ratios Satellites Studies Terrain Theory of Computation |
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| Title | A finite dominating set of cardinality O(k) and a witness set of cardinality O(n) for 1.5D terrain guarding problem |
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