On the traveling salesman problem with neighborhoods

1
On the traveling salesman problem with
neighborhoods

• With
• Khaled Elbassioni
• Aleksei Fishkin
• Hans Bodlaender
• Corinne Feremans
• Alexander Grigoriev
• Eelko Penninkx
• Thomas Wolle

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Euclidean Group TSP
Given n sets of points in the plane find a tour
of minimum length tour that connects all sets.
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TSP with neighborhoods
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TSP with neighborhoods
VLSI design
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TSP with neighborhoods (discrete)
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TSP with neighborhoods

• Known
• 2-e inapproximability (Safra, Schwarz 2003)
• log n approximation (Mata, Mitchell 1995)

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TSP with neighborhoods

• Known
• 2-e inapproximability (Safra, Schwarz 2003)
• log n approximation (Mata, Mitchell 1995)
• O(1)-approx. for
• fat regions (De Berg et al. 2005)
• fat objects (Elbassioni et al. 2005)
• lines (Dumitrescu, Mitchell 2003)

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TSP with neighborhoods

• Known
• 2-e inapproximability (Safra, Schwarz 2003)
• log n approximation (Mata, Mitchell 1995)
• O(1)-approx. for
• fat regions (De Berg et al. 2005)
• fat objects (Elbassioni et al. 2005)
• lines (Dumitrescu, Mitchell 2003)
• PTAS for unit disks (Dumitrescu, Mitchell 2003)
• PTAS for fat objects (Mitchell 2007)

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Group TSP (3? / 2 -approximation)
Each group contains at most ? points.

• Algorithm
• Solve LP
• Round
• Apply Christofides algorithm on selected
  pointset.

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Group TSP
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Group TSP

• Algorithm (3? / 2 -approximation)
• Solve LP-2
• Apply Christofides algorithm on A.

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Group TSP
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Group TSP
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O(1)-approximation for fat regions
D
Fatness a MinD Area(R?O)/Area(D) D has
centre in region R and boundaries intersect
Examples
disk a 1/4
line a 0
halfspace a 1/2
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O(1)-approximation for fat regions
Packing lemma The length of the shortest path
connecting k a-fat regions in R2 is at least
, where d is the diameter of the smallest
region.
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O(1)-approximation for fat regions
Packing lemma The length of the shortest path
connecting k a-fat regions in R2 is at least
, where d is the diameter of the smallest
region.
Proof

• If the centre of a disk with diameter follows
  the path T , then the total area covered is
• at least times
• at most

?
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O(1)-approximation for fat regions
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O(1)-approximation for fat regions
7
6
1
4
5
2
3
Step 1 Order the regions by their diameter
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O(1)-approximation for fat regions
1
Step 1 Order the regions by their diameter
Step 2 In each region pick the point that is
nearest to the already chosen points.
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O(1)-approximation for fat regions
2
Step 1 Order the regions by their diameter
Step 2 In each region pick the point that is
nearest to the already chosen points.
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O(1)-approximation for fat regions
3
Step 1 Order the regions by their diameter
Step 2 In each region pick the point that is
nearest to the already chosen points.
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O(1)-approximation for fat regions
4
Step 1 Order the regions by their diameter
Step 2 In each region pick the point that is
nearest to the already chosen points.
23
O(1)-approximation for fat regions
5
Step 1 Order the regions by their diameter
Step 2 In each region pick the point that is
nearest to the already chosen points.
24
O(1)-approximation for fat regions
6
Step 1 Order the regions by their diameter
Step 2 In each region pick the point that is
nearest to the already chosen points.
25
O(1)-approximation for fat regions
7
Step 1 Order the regions by their diameter
Step 2 In each region pick the point that is
nearest to the already chosen points.
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O(1)-approximation for fat regions
Step 1 Order the regions by their diameter
Step 2 In each region pick the point that is
nearest to the already chosen points.
Step 3 Find a minimum tour on chosen points.
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Analysis
How does it compare with the optimal TSP tour?
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Analysis
How does it compare with the optimal TSP tour?
region j
Path starts at region j and visits the next
k regions in OPT.
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Analysis
The algorithms cost.
xj
region j
xj Connection cost of region j.
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Analysis
Packing lemma The length of the shortest path
connecting k a-fat regions in R2 is at least
, where d is the diameter of the smallest
region.
If k 3/a then T_j is at least the smallest
diameter on the path T_j.
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Analysis
Packing lemma The length of the shortest path
connecting k a-fat regions in R2 is at least
, where d is the diameter of the smallest
region.
If k 3/a then T_j is at least the smallest
diameter on the path T_j.
Consider some region j. Let region h have
smallest diameter on path T_j..
h
If hj then
j
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Analysis
Packing lemma The length of the shortest path
connecting k a-fat regions in R2 is at least
, where d is the diameter of the smallest
region.
If k 3/a then T_j is at least the smallest
diameter on the path T_j.
Consider some region j. Let region h have
smallest diameter on path T_j..
h
If h j, then
j
Otherwise,
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Analysis
Let be set of regions j for which
.
Length of constructed tree is at most
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Analysis
Let be set of regions j for which
. Let be set of regions j for which

Length of constructed tree is at most
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Analysis
Let be set of regions j for which
. Let be set of regions j for which

Length of constructed tree is at most
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Analysis
Let be set of regions j for which
. Let be set of regions j for which

Length of constructed tree is at most
?
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Higher dimensions
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Intersecting fat regions (discrete)
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Intersecting fat regions (discrete)
TSP with neighborhoods (unit disks).
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Intersecting fat regions (discrete)
TSP with neighborhoods (unit disks).
Algorithm Dumitrescu and Mitchell 2003 - Find
maximal independent set. - Connect through walk
over bounderies.
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Intersecting fat regions (discrete)
TSP with neighborhoods (unit disks).
Algorithm Dumitrescu and Mitchell 2003 - Find
maximal independent set. - Connect through walk
over bounderies.
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Intersecting fat regions (discrete)
Algorithm B
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Intersecting fat regions (discrete)
Algorithm B
Step 1 Compute minimum covering box X.
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Intersecting fat regions (discrete)
Algorithm B
Step 1 Compute minimum covering box X.
Step 2 Find minimal hitting set P inside X.
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Intersecting fat regions (discrete)
Algorithm B
Step 1 Compute minimum covering box X.
Step 2 Find minimal hitting set P inside X.
Step 3 Compute a (1e)-approximate TSP-tour on
P.
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Minimum corridor connection problem
Open problem CCCG 2000 Given a rectilinear
decomposition of a square, find the minimum tree
along the edges that connects all sections.
room
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Minimum corridor connection problem
Open problem CCCG 2000 Given a rectilinear
decomposition of a square, find the minimum tree
along the edges that connects all sections.
room
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Minimum corridor connection problem
Open problem CCCG 2000 Given a rectilinear
decomposition of a square, find the minimum tree
along the edges that connects all sections.
w(e)
room
More general Given a weighted planar graph,
connect all faces.
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Subexponential time exact algorithm
Lipton-Tarjan planar separator theorem There
exists a subset of size that cuts
the graph roughly in half.
-time algorithm
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Graphs with small branchwidth
Branch decomposition
5
3
5
1
1
6
3
6
4
2
2
4
G
T
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Graphs with small branchwidth
Branch decomposition
5
3
5
1
1
6
3
6
e
4
2
2
4
G
T
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Graphs with small branchwidth
Branch decomposition
5
3
5
1
1
6
3
6
e
4
2
2
4
G
T
Example - 2 vertices in middle set of e. -
Width of decomposition is 2 (maximum over all e
of T). Branchwidth of graph is minimum width over
all decompositions.
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Graphs with small branchwidth
Planar graphs Sphere cut decomposition
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Graphs with small branchwidth
Planar graphs Sphere cut decomposition
Theorem Seymour and Thomas (1994), Dorn et al.
(2005) If a planar graph has branchwidth B,
then a sphere cut branch decomposition of width B
can be found in O(n3) time.
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Graphs with small branchwidth
Planar graphs Sphere cut decomposition
Theorem Seymour and Thomas (1994), Dorn et al.
(2005) If a planar graph has branchwidth B,
then a sphere cut branch decomposition of width B
can be found in O(n3) time.
Theorem Given a planar graph of branch width B,
the minimum corridor connection problem can be
solved in O(n3 2O(B)) time.
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Graphs with small branchwidth

• For each edge e and triple (S,R,X) we store the
  optimal cost of subproblem.
• S is a subset of the middle set of e
• R is equivalence relation on S
• X is set of faces inside that are connected.
• Size of table for one edge 2O(B)

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Graphs with small branchwidth

• For each edge e and triple (S,R,X) we store the
  optimal cost of subproblem.
• S is a subset of the middle set of e
• R is equivalence relation on S
• faces inside that are connected.
• Size of table for one edge 2O(B)

Given a sphere cut branch decomposition of width
B, the optimal tree can be found by DP in time
2O(B).
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Graphs with small branchwidth
Fact k-outerplanar graphs have branchwidth at
most 2k.
Corollary The mcc can be solved in O(n32O(k))
time on k-outerplanar graphs
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PTAS for equal sized rooms
room
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PTAS for Euclidean TSP (Arora)

• round the instance- build quad tree- restrict
  the instance futher by defining portals on the
  dissection lines- apply DP to restricted
  instance.

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PTAS for Euclidean TSP (Arora)

• round the instance- build quad tree- restrict
  the instance further by defining portals on the
  dissection lines- apply DP to restricted
  instance.

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PTAS for Euclidean TSP (Arora)

• round the instance- build quad tree- restrict
  the instance further by defining portals on the
  dissection lines- apply DP to restricted
  instance.

level 0
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PTAS for Euclidean TSP (Arora)

• round the instance- build quad tree- restrict
  the instance further by defining portals on the
  dissection lines- apply DP to restricted
  instance.

level 0
level 1
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PTAS for Euclidean TSP (Arora)

• round the instance- build quad tree- restrict
  the instance further by defining portals on the
  dissection lines- apply DP to restricted
  instance.

level 0
level 1
level 2
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PTAS for Euclidean TSP

• round the instance- build quad tree- restrict
  the instance further by defining portals on the
  dissection lines- apply DP to restricted
  instance.

level 0
level 1
level 2
level 3
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PTAS for Euclidean TSP

• round the instance- build quad tree- restrict
  the instance further by defining portals on the
  dissection lines- apply DP to restricted
  instance.

m portals per side, mO((log n) / c)
use at most r portals per side for crossing. r
O(1/c)
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PTAS for Euclidean TSP

• round the instance- build quad tree- restrict
  the instance further by defining portals on the
  dissection lines- apply DP to restricted
  instance.

m portals per side, mO((log n) / c)
use at most r portals per side for crossing. r
O(1/c)
size of table for one node mO(r)f(r)
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PTAS for Euclidean TSP

• round the instance- build quad tree- restrict
  the instance further by defining portals on the
  dissection lines- apply DP to restricted
  instance.

m portals per side, mO((log n) / c)
use at most r portals per side for crossing. r
O(1/c)
size of table for one node mO(r)f(r)
time to compute one entry (mO(r)f(r))4 ?
running time n (logn)O(1/c)
nodes in tree n log n
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PTAS for equal sized rooms
Adjusting Aroras algorithm to the mcc problem
Assumption - Each room contains q x q square,
- Perimeter is at most tq for some constant
t 4.
Changes - dissection curves i.o. lines
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PTAS for equal sized rooms
Adjusting Aroras algorithm to the mcc problem
Assumption - Each room contains q x q square,
- Perimeter is at most tq for some constant
t 4.
Changes - dissection curves i.o. lines
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PTAS for equal sized rooms
Adjusting Aroras algorithm to the mcc problem
Assumption - Each room contains q x q square,
- Perimeter is at most tq for some constant
t 4.
Problems - dissection curves i.o. lines
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PTAS for equal sized rooms
Adjusting Aroras algorithm to the mcc problem
Assumption - Each room contains q x q square,
- Perimeter is at most tq for some constant
t 4.
Problems - dissection curves i.o. lines
- crossing paths i.o. points
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PTAS for equal sized rooms
Adjusting Aroras algorithm to the mcc problem
Assumption - Each room contains q x q square,
- Perimeter is at most tq for some constant
t 4.
Problems - dissection curves i.o. lines
- crossing paths i.o. points
portal
- solution follows dissection curves
dis. curve
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PTAS for fat regions (Mitchell 2007)
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PTAS for fat regions
Step 1 Rounding Step 2 D.P.
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PTAS for fat regions

• Step 1 Rounding
• Lines n / e
• Error
• O(n L e / n) O(OPT/e)

L
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PTAS for fat regions
Step 2 D.P.
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PTAS for fat regions
Step 2 D.P. Fix mO(1/ e)
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PTAS for fat regions
Step 2 D.P. Fix mO(1/ e)
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PTAS for Euclidean TSP (Mitchell)
m 2

• Theorem Any set of segments L can be extended to
  a set L such that
• L is m-guillotine and
• L (12v2 / m) L .

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PTAS for Euclidean TSP (Mitchell)

• Definitions
• a point is on the m-span of a line ? if
• a point is m-dark w.r.t. a line ? if ......

?

• Lemma
• There is a line ? for which m-span?
  m-dark? .

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PTAS for Euclidean TSP (Mitchell)
Charged for from below
Each point is charged at most once from each
side. Each segment is charged at most 2v2 / m
times its length.
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PTAS for Euclidean TSP (Mitchell)

• sub problems
• - O(n4) boxes
• 2m points per side
• f(m) ways to combine gt O(n4n16m)
  subproblems.

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PTAS for fat regions
M-region span

• Lemma
• There is a line ? s.t.
• m-span? M-region span? m-dark?
  M-region dark?.

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PTAS for fat regions

• Theorem Any set of segments L can be extended to
  a set L such that
• L is (m,M)-guillotine and
• L (12v2 / m )L C0 / M .

C0 O(D1D2 ... Dn).
(Di is diameter of region i).
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PTAS for fat regions
Lemma OPT O(D1D2 ... Dn-1) / log n.

• M log n / e , m 1/e , C0O(log n OPT) ,
  LOPT
• L (12v2 / m )L C0 / M
• (12v2 e)OPT log n OPT / (log n / e)
• OPT 2v2 e OPT e OPT.

subproblems O(n4n64m) 2(8M) cn O(1/ e)
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Some open problems
MCC for planar graphs Find O(1)-approximation.
TSP on line segments Find O(1)-approximation.
TSP for non-fat regions Find
O(1)-approximation.
Group TSP Find O(log n)-approximation.