this paper, we show that Euclidean TSP has a PTAS. For every fixed c > 1, a randomized version of this Approximation Schemes for Euclidean TSP and other Geometric Problems 3 algorithm computes a (1 + 1/c)-approximation to the optimal tour in O(n(log n) O(c) ) time. When the nodes are in # d , the running time rises to O(n(log n) (O( # dc)) d-1 ). Our algorithm can be derandomized, but this seems to multiply the running time by a factor O(n d ) in # d . Our techniques also apply to many other geometric problems, which are described in Section 1.1
|
868
|
Reducibility among combinatorial problems
– Karp
- 1972
|
|
471
|
Approximation algorithms for combinatorial problems. Proceedings of the fifth annual ACM symposium on Theory of computing 38–49
– Johnson
- 1973
|
|
464
|
Optimization, approximation, and complexity classes
– Papadimitriou, Yannakakis
- 1991
|
|
404
|
Approximation Algorithms for NP-Hard Problems
– Hochbaum
- 1995
|
|
290
|
Probabilistic checking of proofs: a new characterization of NP
– Arora, Safra
- 1992
|
|
258
|
The Traveling Salesman Problem
– Lawler, Lenstra, et al.
- 1985
|
|
236
|
An effective heuristic algorithm for the traveling salesman problem
– Lin, Kernighan
- 1973
|
|
232
|
Bounds for certain multiprocessing anomalies
– Graham
- 1966
|
|
208
|
The traveling salesman problem: A case study in local optimization
– Johnson, McGeoch
- 1996
|
|
195
|
Proof verification and intractability of approximation problems
– Arora, Lund, et al.
- 1998
|
|
195
|
Computer solutions of the traveling salesman problem
– Lin
- 1965
|
|
192
|
TSPLIB { A traveling salesman problem library
– Reinelt
- 1991
|
|
189
|
T.Gonzalez. P-complete approximation problems
– Sahni
- 1976
|
|
184
|
Extensions of Lipschitz mapping into Hilbert space
– Johnson, Lindenstrauss
- 1984
|
|
179
|
Worst-case analysis of a new heuristic for the travelling salesman problem
– Christofides
- 1976
|
|
172
|
Approximating clique is almost NP-complete
– Feige, Goldwasser, et al.
- 1996
|
|
144
|
How easy is local search
– Johnson, Papadimitriou, et al.
- 1988
|
|
133
|
Fast approximation algorithms for the knapsack and sum of subset problems
– Ibarra, Kim
- 1975
|
|
125
|
Guillotine Subdivisions Approximate Polygonal Subdivisions: Part II - A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems
– Mitchell
- 1999
|
|
112
|
Fast algorithms for geometric traveling salesman problems
– Bentley
- 1992
|
|
106
|
Steiner Minimal Trees
– Gilbert, Pollak
- 1968
|
|
96
|
The shortest path through many points
– Beardwood, Halton, et al.
- 1959
|
|
89
|
Approximation schemes for Euclidean k-median and related problems
– Arora, Raghavan, et al.
- 1998
|
|
85
|
Polynomial-time Approximation Schemes for Euclidean TSP and other Geometric Problems
– Arora
- 1998
|
|
84
|
An 11/6-approximation Algorithm for the network Steiner Problem
– Zelikovsky
- 1993
|
|
83
|
An Efficient Approximation Scheme For The One-Dimensional Bin Packing Problem
– Karmakar, Karp
- 1982
|
|
74
|
The complexity of facets (and some facets of complexity
– PAPADIMITRIOU, YANNAKAKIS
- 1984
|
|
71
|
Approximation Algorithms for Geometric Problems
– Bern, Eppstein
- 1995
|
|
70
|
Probabilistic analysis of partitioning algorithms for the traveling-salesman problem in the plane
– Karp
- 1977
|
|
59
|
A 3-approximation for the minimum tree spanning k vertices
– Garg
- 1996
|
|
58
|
Approximation Algorithms for Geometric Tour and Network Design Problems
– Mata, Mitchell
- 1995
|
|
53
|
Some NP-Complete Geometric Problems
– Garey, Graham, et al.
- 1976
|
|
52
|
The Steiner problems with edge lengths 1 and 2
– Bern, Plassmann
- 1989
|
|
49
|
Iterated nearest neighbors and finding minimal polytopes
– Eppstein, Erickson
- 1994
|
|
44
|
Finding cuts in the TSP (a preliminary report
– Applegate, Bixby, et al.
- 1995
|
|
44
|
Geometry helps in matching
– Vaidya
- 1989
|
|
42
|
Parallel construction of quadtrees and quality triangulations
– Bern, Eppstein, et al.
- 1993
|
|
37
|
An approximation scheme for planar graph TSP
– Grigni, Koutsoupias, et al.
- 1995
|
|
37
|
Better approximation bounds for the network and Euclidean Steiner tree problems
– Zelikovsky
- 1995
|
|
34
|
A polynomial-time approximation scheme for weighted palnar graph TSP
– Arora, Grigni, et al.
- 1998
|
|
33
|
A constant-factor approximation for the k-MST problem
– Blum, Chalasani, et al.
- 1995
|
|
33
|
Analyzing the Held-Karp TSP bound: A monotonicity property with application
– Shmoys, Williamson
- 1990
|
|
33
|
Heuristic analysis, linear programming and branch
– Wolsey
- 1980
|
|
32
|
A proof of the Gilbert-Pollak conjecture on the Steiner ratio
– Du, Hwang
- 1992
|
|
30
|
When Hamming meets Euclid: The approximability of geometric TSP
– Trevisan
- 1997
|
|
28
|
Low Degree Spanning Trees of Small Weight
– Khuller, Raghavachari, et al.
- 1996
|
|
18
|
Euclidean TSP is NP-complete
– Papadimitriou
- 1977
|
|
17
|
The complexity of the lin-kernighan heuristic for the traveling salesman problem
– Papadimitriou
- 1992
|
|
16
|
Worst-case Comparison of Valid Inequalities for the TSP
– Goemans
- 1995
|
|
16
|
On the Problem of
– Melzak
- 1961
|
|
