# On the traveling salesman problem with neighborhoods

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```					On the traveling salesman problem
with neighborhoods

With:
Khaled Elbassioni
Aleksei Fishkin
Hans Bodlaender
Corinne Feremans
Alexander Grigoriev
Eelko Penninkx
Thomas Wolle
Euclidean Group TSP

Given n sets of points in the plane find a tour of minimum length tour that
connects all sets.
TSP with neighborhoods
TSP with neighborhoods

VLSI design
TSP with neighborhoods (discrete)
TSP with neighborhoods
Known:
-2-ε inapproximability (Safra, Schwarz 2003)

- log n approximation   (Mata, Mitchell 1995)
TSP with neighborhoods
Known:
-2-ε 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)
TSP with neighborhoods
Known:
-2-ε 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)
Group TSP (3ρ / 2 -approximation)

Each group contains at most ρ points.

Algorithm:
- Solve LP
- Round
- Apply Christofides’ algorithm on selected pointset.
Group TSP
Z1    min    c x
eE
e e                                           Z2    min    c x
eE
e e

s.t.    x
e ( i )
e    2 , for all i  V                          s.t.    x
e ( i )
e     2 yi   , for all i  V

x
e ( S )
e    2 , for all S  V , S  V , S                     x
e ( S )
e    2 yi   , for all S  V , i  S , 0  S
n
0  xe  1            , for all e  E.
a
i 1
ki iy 1      , for all k  1,2,...,m

y0  1

0  xe  1, 0  yi  1 , for all e  E , i  V .
Group TSP
Z1    min    c x
eE
e e                                           Z2    min    c x
eE
e e

s.t.    x
e ( i )
e    2 , for all i  V                          s.t.    x
e ( i )
e     2 yi   , for all i  V

x
e ( S )
e    2 , for all S  V , S  V , S                     x
e ( S )
e    2 yi   , for all S  V , i  S , 0  S
n
0  xe  1            , for all e  E.
a
i 1
ki iy 1      , for all k  1,2,...,m

y0  1

0  xe  1, 0  yi  1 , for all e  E , i  V .

Algorithm: (3ρ / 2 -approximation)
ˆ
- Solve LP-2: Z 2 . Let A  {i V | yi  1 / }
ˆ
- Apply Christofides’ algorithm on A.
Group TSP
Z1     min     c x
eE
e e

s.t.    x
e ( i )
e    2 , for all i  V

x
e ( S )
e    2 , for all S  V , S  V , S  

0  xe  1            , for all e  E.

Z Christofides  3 / 2 Z1
Group TSP
Z1     min      ce xe
W
Z3    min    c x
eE
e e
eE

s.t.    x              2 , for all i  V
e ( i )
e
s.t.    x

e    2 , for all S  V s.t.
e ( S )
S  W   and W \ S  
 xe  2 , for all S  V , S  V , S  
e ( S )
xe  0             , for all e  E.
0  xe  1           , for all e  E.

Z Christofides  3 / 2 Z1                               Z Christofides  3 / 2 ZW
W
3
O(1)-approximation for fat regions

D

Fatness α:
MinD { Area(RO)/Area(D) | D has centre in region R and boundaries intersect}

Examples

disk α =1/4          line α =0       halfspace α =1/2
O(1)-approximation for fat regions

Packing lemma: The length of the shortest path connecting k

α-fat regions in R2 is at least (  ) /4, where δ is the
k   1
diameter of the smallest region.
O(1)-approximation for fat regions

Packing lemma: The length of the shortest path connecting k

α-fat regions in R2 is at least (  ) /4, where δ is the
k   1
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 k times  / 4
             
2
/ 
4 | | k
2
T                 /
4   2

-at most /4 |T|

2
□
O(1)-approximation for fat regions
O(1)-approximation for fat regions

7
6

1
4                 5
2   3

Step 1: Order the regions by their diameter
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
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
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
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
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
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
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
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
Step 3: Find a minimum tour on chosen points.
Analysis
How does it compare with the optimal TSP tour?
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.
Analysis

The algorithms cost.

xj

region j

xj : Connection cost of region j.
Analysis
Packing lemma: The length of the shortest path connecting k

α-fat regions in R2 is at least (  ) /4, where δ is the
k   1
diameter of the smallest region.

If k ≥ 3/α then |T_j| is at least the smallest diameter on the path T_j.
Analysis
Packing lemma: The length of the shortest path connecting k

α-fat regions in R2 is at least (  ) /4, where δ is the
k   1
diameter of the smallest region.

If k ≥ 3/α then |T_j| is at least the smallest diameter on the path T_j.

Consider some region j.
h
Let region h have smallest diameter on path T_j..
If h=j then
j
Analysis
Packing lemma: The length of the shortest path connecting k

α-fat regions in R2 is at least (  ) /4, where δ is the
k   1
diameter of the smallest region.

If k ≥ 3/α then |T_j| is at least the smallest diameter on the path T_j.

Consider some region j.
h
Let region h have smallest diameter on path T_j..
If h = j, then
j
Otherwise,
Analysis
Let   be set of regions j for which      .

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

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

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

Length of constructed tree is at most

□
Higher dimensions
Intersecting fat regions (discrete)
Intersecting fat regions (discrete)

TSP with neighborhoods (unit disks).
Intersecting fat regions (discrete)

TSP with neighborhoods (unit disks).

Algorithm [Dumitrescu and Mitchell 2003]:
- Find maximal independent set.
- Connect through walk over bounderies.
Intersecting fat regions (discrete)

TSP with neighborhoods (unit disks).

Algorithm [Dumitrescu and Mitchell 2003]:
- Find maximal independent set.
- Connect through walk over bounderies.
Intersecting fat regions (discrete)

Algorithm B
Intersecting fat regions (discrete)

Algorithm B
Step 1: Compute minimum covering box X.
Intersecting fat regions (discrete)

Algorithm B
Step 1: Compute minimum covering box X.
Step 2: Find minimal hitting set P’ inside X.
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 (1+e)-approximate TSP-tour on P’.
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
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
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.
Subexponential time exact algorithm

Lipton-Tarjan planar separator theorem:
There exists a subset of size         that cuts the graph roughly in half.

-time algorithm
Graphs with small branchwidth

Branch decomposition

3           5
5       1
1
6
3                                   6
2             4
2
4
G                   T
Graphs with small branchwidth

Branch decomposition

3           5
5       1
1
6
3                                   6
4
e
2                   2
4
G                   T
Graphs with small branchwidth

Branch decomposition

3            5
5                 1
1
6
3                                                     6
4
e
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.
Graphs with small branchwidth

Planar graphs: Sphere cut decomposition
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.
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.
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)
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).
Graphs with small branchwidth

Fact:
k-outerplanar graphs have branchwidth at most 2k.

Corollary:
The mcc can be solved in O(n3+2O(k)) time on k-outerplanar graphs
PTAS for equal sized rooms

room
PTAS for Euclidean TSP (Arora)

- round the instance
- restrict the instance futher by defining
portals on the dissection lines
- apply DP to restricted instance.
PTAS for Euclidean TSP (Arora)

- round the instance
- restrict the instance further by defining
portals on the dissection lines
- apply DP to restricted instance.
PTAS for Euclidean TSP (Arora)

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

level 0
PTAS for Euclidean TSP (Arora)

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

level 0
level 1
PTAS for Euclidean TSP (Arora)

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

level 0
level 1

level 2
PTAS for Euclidean TSP

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

level 0
level 1

level 2

level 3
PTAS for Euclidean TSP

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

level 0                                               m portals per side,
m=O((log n) / c)
level 1

level 2                                     use at most r portals
per side for crossing.
level 3                                          r = O(1/c)
PTAS for Euclidean TSP

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

level 0                                               m portals per side,
m=O((log n) / c)
level 1

level 2                                     use at most r portals
per side for crossing.
level 3                                          r = O(1/c)
size of table for one node: mO(r)f(r)
PTAS for Euclidean TSP

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

level 0                                                 m portals per side,
m=O((log n) / c)
level 1

level 2                                        use at most r portals
per side for crossing.
level 3                                             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
PTAS for equal sized rooms

Adjusting Arora’s 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
PTAS for equal sized rooms

Adjusting Arora’s 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
PTAS for equal sized rooms

Adjusting Arora’s 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
PTAS for equal sized rooms

Adjusting Arora’s 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
PTAS for equal sized rooms

Adjusting Arora’s 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
PTAS for fat regions (Mitchell 2007)
PTAS for fat regions
Step 1: Rounding
Step 2: D.P.
PTAS for fat regions
Step 1: Rounding
# Lines = n / ε

Error:
O(n L ε / n) ≤ O(OPT/ε)

L
PTAS for fat regions
Step 2: D.P.
PTAS for fat regions
Step 2: D.P.

Fix m=O(1/ ε)
PTAS for fat regions
Step 2: D.P.

Fix m=O(1/ ε)
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*| ≤ (1+2√2 / m) |L| .
PTAS for Euclidean TSP (Mitchell)
Definitions
-a point is on the m-span of a line γ if ……
-a point is m-dark w.r.t. a line γ if ......

γ

Lemma
- There is a line γ for which | m-spanγ |≤ | m-darkγ |.
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 2√2 / m times its length.
PTAS for Euclidean TSP (Mitchell)

# sub problems:
- O(n4) boxes
- 2m points per side
- f(m) ways to combine   =>   O(n4n16m) subproblems.
PTAS for fat regions
M-region span

Lemma
- There is a line γ s.t.
|m-spanγ | + |M-region spanγ| ≤ |m-darkγ| + |M-region darkγ|.
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*| ≤ (1+2√2 / m )|L| + C0 / M .

C0 = O(D1+D2+ ... + Dn).

(Di is diameter of region i).
PTAS for fat regions
Lemma OPT = Ω(D1+D2+ ... + Dn-1) / log n.

M = log n / ε , m = 1/ε , C0=O(log n OPT) , L=OPT

|L*| ≤ (1+2√2 / m )|L| + C0 / M
= (1+2√2 ε)OPT + log n OPT / (log n / ε)
= OPT + 2√2 ε OPT + ε OPT.

# subproblems:
O(n4n64m) · 2(8M) = c·n O(1/ ε)
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.

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