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

Glencora Borradaile, David Eppstein
2015 Computational geometry  
We describe 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.  ... 
doi:10.1016/j.comgeo.2015.04.005 fatcat:ysdailf5xvc6pl7la56b4pw6bi

A subset spanner for Planar graphs,

Philip N. Klein
2006 Proceedings of the thirty-eighth annual ACM symposium on Theory of computing - STOC '06  
As a consequence, there is 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.  ... 
doi:10.1145/1132516.1132620 dblp:conf/stoc/Klein06 fatcat:2cjsn6tvx5fsbj7dhc33lklp4i

Approximate TSP in Graphs with Forbidden Minors [chapter]

Michelangelo Grigni
2000 Lecture Notes in Computer Science  
These results imply (i) a QPTAS (quasi-polynomial time 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  ... 
doi:10.1007/3-540-45022-x_73 fatcat:s3un4gipwfevhfyho2rfho4iiy

The Chinese deliveryman problem

Martijn van Ee, René Sitters
2019 4OR  
We give a simple √ 2-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 αapproximation scheme for the CDP.  ... 
doi:10.1007/s10288-019-00420-2 fatcat:hh25yrqpqvfsxbpqdpyazynn6e

Polynomial Time Approximation Scheme for the Minimum-weight k-Size Cycle Cover Problem in Euclidean space of an arbitrary fixed dimension

Michael Khachay, Katherine Neznakhina
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.  ... 
doi:10.1016/j.ifacol.2016.07.541 fatcat:jzsopddbcfb4zjne2ij4lnpsha

Good triangulations yield good tours

Adam N. Letchford, Nicholas A. Pearson
2008 Computers & Operations Research  
Consider the following heuristic 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  ... 
doi:10.1016/j.cor.2006.03.025 fatcat:q7zovspt75clvjhq7wazcsoekq

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.  ... 

Improved lower and upper bounds for universal TSP in planar metrics

Mohammad T. Hajiaghayi, Robert Kleinberg, Tom Leighton
2006 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA '06  
Generalizing from the n×n grid to arbitrary weighted 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  ... 
doi:10.1145/1109557.1109628 fatcat:gdd2kabonjfljn7vamanpd6ria

Algorithms for Optimization Problems in Planar Graphs (Dagstuhl Seminar 16221)

Jeff Erickson, Philip N. Klein, Dániel Marx, Claire Mathieu, Marc Herbstritt
2016 Dagstuhl Reports  
This report documents the program and the outcomes of Dagstuhl Seminar 16221 "Algorithms 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  ... 
doi:10.4230/dagrep.6.5.94 dblp:journals/dagstuhl-reports/EricksonKMM16 fatcat:wasdfgivt5fqdppfxo3iqqs2ta

Polynomial time approximation schemes for the traveling repairman and other minimum latency problems [chapter]

René Sitters
2013 Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms  
We give a polynomial time, (1 + )-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.  ... 
doi:10.1137/1.9781611973402.46 dblp:conf/soda/Sitters14 fatcat:tnyitwpusvcdvgkpd6cqvsbhji

A Linear-Time Approximation Scheme for TSP in Undirected Planar Graphs with Edge-Weights

Philip N. Klein
2008 SIAM journal on computing (Print)  
We give 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.  ... 
doi:10.1137/060649562 fatcat:qkvmtagkdvgrbobjn2zi2fgsku

Approximation Schemes for Planar Graph Problems [chapter]

Erik D. Demaine, MohammadTaghi Hajiaghayi
2008 Encyclopedia of Algorithms  
This problem is inapproximable in general 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.  ... 
doi:10.1007/978-0-387-30162-4_32 fatcat:kad2i2t5gres3oua25e3xklc3u

A near-linear time minimum Steiner cut algorithm for planar graphs [article]

Stephen Jue, Philip N. Klein
2019 arXiv   pre-print
We consider the Minimum Steiner Cut problem on undirected 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.  ... 
arXiv:1912.11103v2 fatcat:nxakl2fgb5bwje46wg6xwadzpq

The Traveling Salesman Problem under Squared Euclidean Distances

Fred Van Nijnatten, René Sitters, Gerhard J. Woeginger, Alexander Wolff, Mark De Berg, Marc Herbstritt
2010 Symposium on Theoretical Aspects of Computer Science  
We design a 5-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] .  ... 
doi:10.4230/lipics.stacs.2010.2458 dblp:conf/stacs/NijnattenSWWB10 fatcat:kbwl6mbmtjf2ziqr2xts575mju

The Traveling Salesman Problem Under Squared Euclidean Distances [article]

Mark de Berg and Fred van Nijnatten and René Sitters and Gerhard J. Woeginger and Alexander Wolff
2010 arXiv   pre-print
We design a 5-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] .  ... 
arXiv:1001.0236v3 fatcat:dk2ic7bm2rhalgzsfv5bwfnzwe
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