## Resource-Constrained Geometric Network Optimization (Extended Abstract)

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Citations: | 22 - 1 self |

### BibTeX

@MISC{Arkin_resource-constrainedgeometric,

author = {Esther M. Arkin and Joseph S. B. Mitchell and Giri Narasimhan},

title = {Resource-Constrained Geometric Network Optimization (Extended Abstract) },

year = {}

}

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### Abstract

We study a variety of geometric network optimization problems on a set of points, in which we are given a resource bound, B, on the total length of the network, and our objective is to maximize the number of points visited (or the total "value" of points visited). In particular, we resolve the well-publicized open problem on the approximability of the rooted "orienteering problem" for the case in which the sites are given as points in the plane and the network required is a cycle. We obtain a 2-approximation for this problem. We also obtain approximation algorithms for variants of this problem in which the network required is a tree (3-approximation) or a path (2-approximation). No prior approximation bounds were known for any of these problems. We also obtain improved approximation algorithms for geometric instances of the unrooted orienteering problem, where we obtain a 2-approximation for both the cycle and tree versions of the problem on points in the plane, as well as a ...

### Citations

347 | A general approximation technique for constrainedforestproblems
- Goemans, Williamson
- 1995
(Show Context)
Citation Context ...discussed in [7], an approximation algorithm follows from concatenating a cycle obtained for the quota-driven salesman, with the 2-approximation cycle given by the algorithm of Goemans and Williamson =-=[15]-=- (which considers the effect of penalties, but does not use the quota constraint). The class of problems we address in this paper is based on the orienteering problem (or bank robber problem). The pro... |

318 | Polynomial Time Approximation Schemes for Euclidean TSP and other Geometric Problems
- Arora
- 1996
(Show Context)
Citation Context ...l case of undirected networks whose edge lengths obey the triangle inequality. Polynomial-time approximation schemes (PTAS) for geometric (e.g., Euclidean) instances were discovered recently by Arora =-=[4, 5] and -=-by Mitchell [21, 23, 24]; most recently, Rao and Smith [27] give a deterministic PTAS with running time O(n log n) in any fixed dimension. The methods of [21, 23] employ "m-guillotine subdivision... |

313 |
The Traveling Salesman Problem
- Lawler
- 1985
(Show Context)
Citation Context ...ell as in analyzing the generalizations to non-point sites. Related Work There has been a wealth of work on the related traveling salesperson problem (TSP), both in networks and in geometric settings =-=[9, 18, 29, 17]-=-. TSP is NP-hard, even for points in the Euclidean plane. Until recently, the best approximation algorithm known for the Euclidean TSP was the Christofides heuristic (see, e.g., [18]), which yields a ... |

164 | Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems
- Mitchell
- 1999
(Show Context)
Citation Context ...tworks whose edge lengths obey the triangle inequality. Polynomial-time approximation schemes (PTAS) for geometric (e.g., Euclidean) instances were discovered recently by Arora [4, 5] and by Mitchell =-=[21, 23, 24]; most rec-=-ently, Rao and Smith [27] give a deterministic PTAS with running time O(n log n) in any fixed dimension. The methods of [21, 23] employ "m-guillotine subdivisions" in the plane; this tool is... |

147 | Geometric shortest paths and network optimization
- Mitchell
- 2000
(Show Context)
Citation Context ...s in bicriteria path optimization, in which we are to minimize one measure of "length", subject to an upper bound on another notion of "length"; see [2, 3, 11, 25], as well as the =-=surveys by Mitchell [20, 22]-=-. Preliminaries We assume that the input to our problems is given by a set of n points (sites) in the Euclidean plane. For the path and cycle cases, we also consider a discrete metric space (an edgewe... |

129 |
Fast algorithms for geometric traveling salesman problems
- Bentley
- 1992
(Show Context)
Citation Context ...ell as in analyzing the generalizations to non-point sites. Related Work There has been a wealth of work on the related traveling salesperson problem (TSP), both in networks and in geometric settings =-=[9, 18, 29, 17]-=-. TSP is NP-hard, even for points in the Euclidean plane. Until recently, the best approximation algorithm known for the Euclidean TSP was the Christofides heuristic (see, e.g., [18]), which yields a ... |

113 |
The traveling salesman problem
- Junguer, Reinelt, et al.
- 1995
(Show Context)
Citation Context ...ell as in analyzing the generalizations to non-point sites. Related Work There has been a wealth of work on the related traveling salesperson problem (TSP), both in networks and in geometric settings =-=[9, 18, 29, 17]-=-. TSP is NP-hard, even for points in the Euclidean plane. Until recently, the best approximation algorithm known for the Euclidean TSP was the Christofides heuristic (see, e.g., [18]), which yields a ... |

97 |
The complexity of computing Steiner minimal trees
- Garey, Graham, et al.
(Show Context)
Citation Context ...hods, we again restrict ourselves to the Euclidean plane. The multiply-rooted tree-orienteering problem is closely related to the (planar) Euclidean Steiner tree problem, which is known to be NP-hard =-=[13]-=-. In the Steiner tree problem, one is to determine a tree of minimum total length whose vertices are a superset of a given set of points. The hardness of the Steiner tree problem implies that it is NP... |

93 |
The prize collecting traveling salesman problem
- BALAS
- 1989
(Show Context)
Citation Context ...isited is at least R. By replicating each site (w i times), a k-MST approximation algorithm also gives an approximation for this problem. In the prize-collecting salesman problem, as studied by Balas =-=[8] (see also-=- [10]), the setup is the same as in the quota-driven salesman problem, except that, in addition to "values" w i , there are non-negative penalties associated with each site, and now the obje... |

91 | Nearly linear time approximation schemes for Euclidean TSP and other geometric problems
- Arora
- 1998
(Show Context)
Citation Context ...l case of undirected networks whose edge lengths obey the triangle inequality. Polynomial-time approximation schemes (PTAS) for geometric (e.g., Euclidean) instances were discovered recently by Arora =-=[4, 5] and -=-by Mitchell [21, 23, 24]; most recently, Rao and Smith [27] give a deterministic PTAS with running time O(n log n) in any fixed dimension. The methods of [21, 23] employ "m-guillotine subdivision... |

78 |
A 3-approximation for the minimum tree spanning k vertices
- Garg
- 1996
(Show Context)
Citation Context ... problem is known to be NP-hard, both in general graphs and in the Euclidean plane [12, 28, 31]. The current best approximation algorithm for general edge-weighted graphs is a 3-approximation by Garg =-=[14], which applies also-=- to the "rooted" case (the tree is required to include a given node); this has been improved to a 2.5-approximation, by Arya and Ramesh [6], if the tree is not "rooted." The Euclid... |

76 |
Planar Graphs: Theory and Algorithms
- Nishizeki, Chiba
- 1988
(Show Context)
Citation Context ... types (1) and (2), there must be one whose weight is less than B, while the cardinality, k, of the point set spanned is at least n(T )=3. Proof. By standard results on separators in trees (e.g., see =-=[26]-=-, Theorem 9.1, page 150), we know that there exists a partitioning of the optimal tree T into two subtrees, T1 and T2 , sharing a common vertex v of T , so that (2=3)n(T )sn(T i )s(1=3)n(T ). Without ... |

68 |
Shortest paths and networks
- Mitchell
- 2004
(Show Context)
Citation Context ...s in bicriteria path optimization, in which we are to minimize one measure of "length", subject to an upper bound on another notion of "length"; see [2, 3, 11, 25], as well as the =-=surveys by Mitchell [20, 22]-=-. Preliminaries We assume that the input to our problems is given by a set of n points (sites) in the Euclidean plane. For the path and cycle cases, we also consider a discrete metric space (an edgewe... |

67 | Approximation algorithms for geometric tour and network design problems
- Mata, Mitchell
- 1995
(Show Context)
Citation Context ...btained O(1)- approximation algorithms for "well behaved" (possibly overlapping) regions (e.g., regions are disks or have roughly equal-length and parallel diameter segments), while Mata and=-= Mitchell [19] have obta-=-ined an O(log n)-approximation algorithm for n general (possibly overlapping) regions, based on "guillotine rectangular subdivisions". We prove the following theorem, giving the first approx... |

66 | Spanning trees short or small
- Ravi, Sundaram, et al.
- 1996
(Show Context)
Citation Context ...s, and an integer ksn, and our goal is to find a tree of least total weight that spans some subset of k vertices. The problem is known to be NP-hard, both in general graphs and in the Euclidean plane =-=[12, 28, 31]. The curr-=-ent best approximation algorithm for general edge-weighted graphs is a 3-approximation by Garg [14], which applies also to the "rooted" case (the tree is required to include a given node); t... |

63 | A note on the prize collecting traveling salesman problem
- BIENSTOCK, GOEMANS, et al.
- 1993
(Show Context)
Citation Context ...east R. By replicating each site (w i times), a k-MST approximation algorithm also gives an approximation for this problem. In the prize-collecting salesman problem, as studied by Balas [8] (see also =-=[10]), the set-=-up is the same as in the quota-driven salesman problem, except that, in addition to "values" w i , there are non-negative penalties associated with each site, and now the objective is to min... |

62 | Approximation algorithms for the geometric covering salesman problem
- Arkin, Hassin
- 1994
(Show Context)
Citation Context ...closely related to the "TSP with Neighborhoods" (TSPN) problem. We assume that the regions are given as a collection of n simple polygons, having a total of N vertices. For the TSPN, Arkin a=-=nd Hassin [1] have obta-=-ined O(1)- approximation algorithms for "well behaved" (possibly overlapping) regions (e.g., regions are disks or have roughly equal-length and parallel diameter segments), while Mata and Mi... |

57 |
The orienteering problem
- Golden, Levy, et al.
- 1987
(Show Context)
Citation Context ...ering problem has also been called the "bank robber" problem (a thief wishes to maximize the "haul", with a single tank of gas in the get-away car) and the "generalized travel=-=ing salesperson problem" [16, 30]. The name-=- "orienteering" comes from the outdoor sport by the same name, in which each player is given a compass and a map of a given outdoor terrain, and has the goal to accumulate the largest score ... |

53 | New Approximation Guarantees for Minimum-Weight k-Tress and Prize-Collecting Salesmen
- Awebuch, Azar, et al.
(Show Context)
Citation Context ...ated with each site, and now the objective is to minimize the sum of the distances traveled plus the sum of the penalties on the points not visited, subject to satisfying the quota R. As discussed in =-=[7]-=-, an approximation algorithm follows from concatenating a cycle obtained for the quota-driven salesman, with the 2-approximation cycle given by the algorithm of Goemans and Williamson [15] (which cons... |

27 |
Weighted k-cardinality trees: complexity and polyhedral structure
- Fischetti, Hamacher, et al.
- 1994
(Show Context)
Citation Context ...s, and an integer ksn, and our goal is to find a tree of least total weight that spans some subset of k vertices. The problem is known to be NP-hard, both in general graphs and in the Euclidean plane =-=[12, 28, 31]. The curr-=-ent best approximation algorithm for general edge-weighted graphs is a 3-approximation by Garg [14], which applies also to the "rooted" case (the tree is required to include a given node); t... |

26 |
Fast heuristics for large geometric travelling salesman problems
- Reinelt
- 1992
(Show Context)
Citation Context |

24 |
Heuristic methods applied to orienteering
- Tsiligirides
- 1984
(Show Context)
Citation Context ...ering problem has also been called the "bank robber" problem (a thief wishes to maximize the "haul", with a single tank of gas in the get-away car) and the "generalized travel=-=ing salesperson problem" [16, 30]. The name-=- "orienteering" comes from the outdoor sport by the same name, in which each player is given a compass and a map of a given outdoor terrain, and has the goal to accumulate the largest score ... |

20 | Finding minimum area k-gons
- Eppstein, Overmars, et al.
- 1992
(Show Context)
Citation Context ...tion algorithm. Our work is also related to problems in bicriteria path optimization, in which we are to minimize one measure of "length", subject to an upper bound on another notion of &quo=-=t;length"; see [2, 3, 11, 25]-=-, as well as the surveys by Mitchell [20, 22]. Preliminaries We assume that the input to our problems is given by a set of n points (sites) in the Euclidean plane. For the path and cycle cases, we als... |

20 |
Computing a shortest k-link path in a polygon
- Mitchell, Piatko, et al.
- 1992
(Show Context)
Citation Context ...tion algorithm. Our work is also related to problems in bicriteria path optimization, in which we are to minimize one measure of "length", subject to an upper bound on another notion of &quo=-=t;length"; see [2, 3, 11, 25]-=-, as well as the surveys by Mitchell [20, 22]. Preliminaries We assume that the input to our problems is given by a set of n points (sites) in the Euclidean plane. For the path and cycle cases, we als... |

17 | A 2.5 factor approximation algorithm for the k-MST problem
- Arya, Ramesh
- 1998
(Show Context)
Citation Context ...ted graphs is a 3-approximation by Garg [14], which applies also to the "rooted" case (the tree is required to include a given node); this has been improved to a 2.5-approximation, by Arya a=-=nd Ramesh [6], if the t-=-ree is not "rooted." The Euclidean kMST now has a PTAS [4, 5, 21, 23], as does the k-TSP, which asks for a shortest cycle visiting some subset of k points. For the graph version of the k-TSP... |

11 |
J.S.B.: Geometric knapsack problems
- Arkin, Khuller, et al.
- 1993
(Show Context)
Citation Context ...tion algorithm. Our work is also related to problems in bicriteria path optimization, in which we are to minimize one measure of "length", subject to an upper bound on another notion of &quo=-=t;length"; see [2, 3, 11, 25]-=-, as well as the surveys by Mitchell [20, 22]. Preliminaries We assume that the input to our problems is given by a set of n points (sites) in the Euclidean plane. For the path and cycle cases, we als... |

11 |
Bicriteria shortest path problems in the plane
- Arkin, Mitchell, et al.
- 1991
(Show Context)
Citation Context |

11 |
Minimal and bounded trees
- Zelikovsky, Lozevanu
- 1993
(Show Context)
Citation Context ...s, and an integer ksn, and our goal is to find a tree of least total weight that spans some subset of k vertices. The problem is known to be NP-hard, both in general graphs and in the Euclidean plane =-=[12, 28, 31]. The curr-=-ent best approximation algorithm for general edge-weighted graphs is a 3-approximation by Garg [14], which applies also to the "rooted" case (the tree is required to include a given node); t... |

6 |
Improved approximation schemes for traveling salesman tours
- Rao, Smith
- 1998
(Show Context)
Citation Context ... inequality. Polynomial-time approximation schemes (PTAS) for geometric (e.g., Euclidean) instances were discovered recently by Arora [4, 5] and by Mitchell [21, 23, 24]; most recently, Rao and Smith =-=[27] give a de-=-terministic PTAS with running time O(n log n) in any fixed dimension. The methods of [21, 23] employ "m-guillotine subdivisions" in the plane; this tool is also essential for some of our res... |