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Near-linear-time deterministic plane Steiner spanners for well-spaced point sets

2015
*
Computational geometry
*

We describe

doi:10.1016/j.comgeo.2015.04.005
fatcat:ysdailf5xvc6pl7la56b4pw6bi
*an*algorithm that takes as input n points in the plane and a parameter ϵ, and produces as output*an*embedded*planar**graph*having the given points as a subset of its vertices in which the*graph*... We use this result in a similarly fast deterministic*approximation**scheme**for*the traveling salesperson problem. ...*Approximating*the*TSP**An*algorithm of Klein [15] provides a linear time*approximation**scheme**for*the traveling salesperson problem in a*planar**graph*. ...##
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A subset spanner for Planar graphs,

2006
*
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing - STOC '06
*

As a consequence, there is

doi:10.1145/1132516.1132620
dblp:conf/stoc/Klein06
fatcat:2cjsn6tvx5fsbj7dhc33lklp4i
*an*O(n log n)-time*approximation**scheme**for*finding a*TSP*among a given subset of nodes of a*planar**graph*. This is the first PTAS*for*the problem. ...*For*any edge-weighted*planar**graph*G and a subset S of nodes of G, there is a subgraph H of G of weight a constant times that of the minimum Steiner tree*for*S such that distances in H between nodes in ... Acknowlegements Thanks to Glencora Borradaile and Claire Kenyon*for*helpful discussions. ...##
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Approximate TSP in Graphs with Forbidden Minors
[chapter]

2000
*
Lecture Notes in Computer Science
*

These results imply (i) a QPTAS (quasi-polynomial time

doi:10.1007/3-540-45022-x_73
fatcat:s3un4gipwfevhfyho2rfho4iiy
*approximation**scheme*)*for*the*TSP*(traveling salesperson problem) in unweighted*graphs*with*an*excluded minor, and (ii) a QPTAS*for*the*TSP*in weighted ... Given as input*an*edge-weighted*graph*, we analyze two algorithms*for*nding subgraphs with low total edge weight. ... Acknowledgment: We thank Robin Thomas*for*help with Lemma 10. References ...##
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The Chinese deliveryman problem

2019
*
4OR
*

We give a simple √ 2-

doi:10.1007/s10288-019-00420-2
fatcat:hh25yrqpqvfsxbpqdpyazynn6e
*approximation**for*undirected*graphs*and a 4/3-*approximation**for*2-edge connected*graphs*. ... We observe that there is a PTAS*for**planar**graphs*and that depth first search is optimal*for*trees. ... A PTAS*for**planar**graphs*As observed in Sect. 1.1, any α-*approximation**scheme**for*TRP implies*an*α*approximation**scheme**for*the CDP. ...##
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Polynomial Time Approximation Scheme for the Minimum-weight k-Size Cycle Cover Problem in Euclidean space of an arbitrary fixed dimension

2016
*
IFAC-PapersOnLine
*

*For*the Euclidean Min-k-SCCP in R d , we construct a polynomial-time

*approximation*

*scheme*, which generalizes the approach proposed earlier

*for*the

*planar*Min-2-SCCP. ...

*For*the Euclidean Min-k-SCCP in R d , we construct a polynomial-time

*approximation*

*scheme*, which generalizes the approach proposed earlier

*for*the

*planar*Min-2-SCCP. ...

*For*the Euclidean Min-k-SCCP in R d , we construct a polynomial-time

*approximation*

*scheme*, which generalizes the approach proposed earlier

*for*the

*planar*Min-2-SCCP. ...

##
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Good triangulations yield good tours

2008
*
Computers & Operations Research
*

Consider the following heuristic

doi:10.1016/j.cor.2006.03.025
fatcat:q7zovspt75clvjhq7wazcsoekq
*for**planar*Euclidean instances of the traveling salesman problem (*TSP*): select a subset of the edges which induces a*planar**graph*, and solve either the*TSP*or its graphical ... It turns out that the Delaunay and greedy triangulations make effective choices*for*the induced*planar**graph*. ... The*planar**graph**TSP*, on the other hand, is not regarded as a special case of the metric*TSP*: although*an*instance of the*planar**graph**TSP*can be transformed to a metric*TSP*instance using the standard ...##
###
Page 1329 of Mathematical Reviews Vol. , Issue 2003B
[page]

2003
*
Mathematical Reviews
*

; Kiel)
Polynomial time

*approximation**schemes**for*Max-Bisection on*planar*and geometric*graphs*. ... Better*approximations*are obtained*for**planar**graphs*and*for*cubic*graphs*. ...##
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Improved lower and upper bounds for universal TSP in planar metrics

2006
*
Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA '06
*

Generalizing from the n×n grid to arbitrary weighted

doi:10.1145/1109557.1109628
fatcat:gdd2kabonjfljn7vamanpd6ria
*planar**graph*metrics, and more generally H-minor-free metrics, we improve the best known upper bound*for*universal tours of such metrics from O(log ... Here, we settle this conjecture negatively by proving*an*Ω " 6 p log n/ log log n " lower bound*for*universal*TSP*tours of the n×n grid; this is the first known example of a family of finite metrics with ... Upper Bound*for*Universal*TSP*in*Planar**Graphs*and H-minor-free*Graphs*Our approach to obtain O(log 2 n) competitive ratio*for*universal*TSP*in*planar**graphs*relies on a partitioning technique*for**planar*...##
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Algorithms for Optimization Problems in Planar Graphs (Dagstuhl Seminar 16221)

2016
*
Dagstuhl Reports
*

This report documents the program and the outcomes of Dagstuhl Seminar 16221 "Algorithms

doi:10.4230/dagrep.6.5.94
dblp:journals/dagstuhl-reports/EricksonKMM16
fatcat:wasdfgivt5fqdppfxo3iqqs2ta
*for*Optimization Problems in*Planar**Graphs*". The seminar was held from May 29 to June 3, 2016. ... This report contains abstracts*for*the recent developments in*planar**graph*algorithms discussed during the seminar as well as summaries of open problems in this area of research. ... Paradigms*for*obtaining*approximation**schemes**for**planar**graphs*Research on polynomial-time*approximation**schemes**for*optimization problems in*planar**graphs*goes back to the pioneering work of Lipton ...##
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Polynomial time approximation schemes for the traveling repairman and other minimum latency problems
[chapter]

2013
*
Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms
*

We give a polynomial time, (1 + )-

doi:10.1137/1.9781611973402.46
dblp:conf/soda/Sitters14
fatcat:tnyitwpusvcdvgkpd6cqvsbhji
*approximation*algorithm*for*the traveling repairman problem (TRP) in the Euclidean plane, on weighted*planar**graphs*, and on weighted trees. ... Subsequently,*approximation**schemes*are given*for*this segmented*TSP*problem in different metric spaces. ...*Planar**graph*A PTAS*for*the*TSP*on weighted*planar**graphs*was given by Arora et al. [9] . The separator in this case is a Jordan curves that divides the*graph*into*an*exterior and interior part. ...##
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A Linear-Time Approximation Scheme for TSP in Undirected Planar Graphs with Edge-Weights

2008
*
SIAM journal on computing (Print)
*

We give

doi:10.1137/060649562
fatcat:qkvmtagkdvgrbobjn2zi2fgsku
*an*algorithm requiring O(c 1/ǫ 2 n) time to find*an*ǫ-optimal traveling salesman tour in the shortest-path metric defined by*an*undirected*planar**graph*with nonnegative edgelengths. ...*For*the case of all lengths equal to 1, the time required is O(c 1/ǫ n). ... Thanks also to Glencora Borradaile and Erik Demaine*for*helpful discussions and suggestions. ...##
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Approximation Schemes for Planar Graph Problems
[chapter]

2008
*
Encyclopedia of Algorithms
*

This problem is inapproximable in general

doi:10.1007/978-0-387-30162-4_32
fatcat:kad2i2t5gres3oua25e3xklc3u
*graphs*within log n*for*some > 0 unless P = NP, but as we will see,*for**planar**graphs*the problem admits a polynomial-time*approximation**scheme*(PTAS): a collection ... This problem is inapproximable in general*graphs*within a factor of n 1−*for*any > 0 unless NP = ZPP (and inapproximable within n 1/2− unless P = NP), while*for**planar**graphs*there is a 4-*approximation*... A polynomial-time*approximation**scheme**for*Steiner tree in*planar**graphs*. In Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, 2007. ...##
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A near-linear time minimum Steiner cut algorithm for planar graphs
[article]

2019
*
arXiv
*
pre-print

We consider the Minimum Steiner Cut problem on undirected

arXiv:1912.11103v2
fatcat:nxakl2fgb5bwje46wg6xwadzpq
*planar**graphs*with non-negative edge weights. ... Steiner (Subset)*TSP*. ... We are now implementing*an**approximation**scheme*[14]*for*Steiner*TSP*in edge-weighted*planar**graphs*. 4*For*Steiner*TSP*, the Held-Karp lower bound and subtour elimination lower bound are not valid. ...##
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The Traveling Salesman Problem under Squared Euclidean Distances

2010
*
Symposium on Theoretical Aspects of Computer Science
*

We design a 5-

doi:10.4230/lipics.stacs.2010.2458
dblp:conf/stacs/NijnattenSWWB10
fatcat:kbwl6mbmtjf2ziqr2xts575mju
*approximation*algorithm*for**Tsp*(2,2) and generalize this result to obtain*an**approximation*factor of 3 α−1 + √ 6 α /3*for*d = 2 and all α 2. ... We present a polynomial-time*approximation**scheme**for*Rev-*Tsp*(2, α) with α 2, and we show that Rev-*Tsp*(d, α) is apx-hard if d 3 and α > 1. ... The Gabriel*graph*is*planar*. Hence we end up with*an*instance of the*Tsp*on weighted*planar**graphs*,*for*which a PTAS is known [5] . ...##
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The Traveling Salesman Problem Under Squared Euclidean Distances
[article]

2010
*
arXiv
*
pre-print

We design a 5-

arXiv:1001.0236v3
fatcat:dk2ic7bm2rhalgzsfv5bwfnzwe
*approximation*algorithm*for**TSP*(2,2) and generalize this result to obtain*an**approximation*factor of 3^α-1+√(6)^α/3*for*d=2 and all α>2. ... We present a polynomial-time*approximation**scheme**for*Rev-*TSP*(2,α) with α>2, and we show that Rev-*TSP*(d, α) is APX-hard if d>3 and α>1. ... The Gabriel*graph*is*planar*. Hence we end up with*an*instance of the*Tsp*on weighted*planar**graphs*,*for*which a PTAS is known [5] . ...
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