Euclidean TSP in Narrow Strips
We investigate how the complexity of Euclidean TSP for point sets P inside the strip ( - ∞ , + ∞ ) × [ 0 , δ ] depends on the strip width δ . We obtain two main results. For the case where the points have distinct integer x -coordinates, we prove that a shortest bitonic tour (which can be computed...
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| Veröffentlicht in: | Discrete & computational geometry Jg. 71; H. 4; S. 1456 - 1506 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
New York
Springer US
01.06.2024
Springer Nature B.V |
| Schlagworte: | |
| ISSN: | 0179-5376, 1432-0444 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | We investigate how the complexity of
Euclidean TSP
for point sets
P
inside the strip
(
-
∞
,
+
∞
)
×
[
0
,
δ
]
depends on the strip width
δ
. We obtain two main results.
For the case where the points have distinct integer
x
-coordinates, we prove that a shortest bitonic tour (which can be computed in
O
(
n
log
2
n
)
time using an existing algorithm) is guaranteed to be a shortest tour overall when
δ
⩽
2
2
, a bound which is best possible.
We present an algorithm that is fixed-parameter tractable with respect to
δ
. Our algorithm has running time
2
O
(
δ
)
n
+
O
(
δ
2
n
2
)
for sparse point sets, where each
1
×
δ
rectangle inside the strip contains
O
(1) points. For random point sets, where the points are chosen uniformly at random from the rectangle
[
0
,
n
]
×
[
0
,
δ
]
, it has an expected running time of
2
O
(
δ
)
n
. These results generalise to point sets
P
inside a hypercylinder of width
δ
. In this case, the factors
2
O
(
δ
)
become
2
O
(
δ
1
-
1
/
d
)
. |
|---|---|
| Bibliographie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0179-5376 1432-0444 |
| DOI: | 10.1007/s00454-023-00609-7 |