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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| Published in: | Annals of operations research Vol. 254; no. 1-2; pp. 37 - 46 |
|---|---|
| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
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Springer US
01.07.2017
Springer Springer Nature B.V |
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| 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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| Keywords | Finite dominating sets Terrain guarding problem Zero-one integer programming Location |
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| References_xml | – volume: 18 start-page: 195 year: 2010 end-page: 210 ident: CR3 article-title: Solving nonconvex planar location problems by finite dominating sets publication-title: Journal of Global Optimization doi: 10.1023/A:1008395305189 – volume: 96 start-page: 191 year: 2000 end-page: 208 ident: CR16 article-title: Planar Weber location problems with barriers and block norms publication-title: Annals of Operations Research doi: 10.1023/A:1018951502447 – volume: 39 start-page: 100 issue: 1 year: 1991 end-page: 118 ident: CR17 article-title: Finite dominating sets for network location problems publication-title: Operations Research doi: 10.1287/opre.39.1.100 – ident: CR18 – volume: 89 start-page: 2143 issue: 16 year: 2012 end-page: 2151 ident: CR11 article-title: Guarding 1.5D terrains with demands publication-title: International Journal of Computer Mathematics doi: 10.1080/00207160.2012.707800 – volume: 36 start-page: 1631 issue: 6 year: 2007 end-page: 1647 ident: CR1 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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
) to cover/guard the whole... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) 1.5 dimensional (1.5D) terrain is characterized by a piecewise linear curve.... |
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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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