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```					   The Travelling Salesman Problem
a brief survey

Martin Grötschel
Summary of Chapter 2
of the class
Polyhedral Combinatorics (ADM III)
May 18, 2010

Martin Grötschel     Institute of Mathematics, Technische Universität Berlin (TUB)
 DFG-Research Center “Mathematics for key technologies” (MATHEON)
 Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)
groetschel@zib.de   http://www.zib.de/groetschel
2

Contents
1. Introduction
2. The TSP and some of its history
3. The TSP and some of its variants
4. Some applications
5. Modeling issues
6. Heuristics
7. How combinatorial optimizers do it

Martin
Grötschel
3

Contents
1. Introduction
2. The TSP and some of its history
3. The TSP and some of its variants
4. Some applications
5. Modeling issues
6. Heuristics
7. How combinatorial optimizers do it

Martin
Grötschel
4

Combinatorial optimization
Given a finite set E and a subset I of the power set of E (the set of
feasible solutions). Given, moreover, a value (cost, length,…) c(e) for all
elements e of E. Find, among all sets in I, a set I such that its total value
c(I) (= sum of the values of all elements in I) is as small (or as large) as
possible.

The parameters of a combinatorial optimization problem are: (E, I, c).

                      
min c(I)   c(e) | I  I  , where I  2E and E finite
       eI            
An important issue: How is I given?

Martin
Grötschel
5

Special „simple“
combinatorial optimization problems
Finding a
 minimum spanning tree in a graph
 shortest path in a directed graph
 maximum matching in a graph
 a minimum capacity cut separating two given nodes of a
graph or digraph
 cost-minimal flow through a network with capacities and
costs on all edges
 …
These problems are solvable in polynomial time.
Martin
Grötschel
6

Special „hard“
combinatorial optimization problems
 travelling salesman problem (the prototype problem)
 location und routing
 set-packing, partitioning, -covering
 max-cut
 linear ordering
 scheduling (with a few exceptions)
 node and edge colouring
 …
These problems are NP-hard
(in the sense of complexity theory).

Martin
Grötschel
7

The travelling salesman problem
Given n „cities“ and „distances“ between them.
Find a tour (roundtrip) through all cities visiting
every city exactly once such that the sum of all
distances travelled is as small as possible. (TSP)

The TSP is called symmetric (STSP) if, for every
pair of cities i and j, the distance from i to j is
the same as the one from j to i, otherwise the
problem is called aysmmetric (ATSP).
Martin
Grötschel
http://www.tsp.gatech.edu/
9

THE TSP
book

suggested reading for
everyone interested
in the TSP

Martin
Grötschel
10

The travelling salesman problem
Two mathematical formulations of the TSP
1. Version :
Let K n  (V , E ) be the complete graph digraph with n nodes
and let ce be the length of e  E. Let H be the set of all
hamiltonian cycles (tours ) in K n . Find
min{c(T ) | T  H }.

2. Version :
Find a cyclic permutation  of {1,..., n} such that
n

c
i 1
i (i )

is as small as possible.

Martin
 Does that help solve the TSP?
Grötschel
11

Contents
1. Introduction
2. The TSP and some of its history
3. The TSP and some of its variants
4. Some applications
5. Modeling issues
6. Heuristics
7. How combinatorial optimizers do it

Martin
Grötschel
12

Usually quoted as
the forerunner of
the TSP

Usually quoted as
the origin of
the TSP

Martin
Grötschel
about 100
years
earlier
14

By a proper choice and
scheduling of the tour one
can gain so much time
that we have to make
some suggestions

The most important
aspect is to cover as many
locations as possible
without visiting a
location twice

Martin
Grötschel
15

Ulysses roundtrip (an even older TSP ?)

The paper „The Optimized Odyssey“ by Martin Grötschel
and Manfred Padberg is downloadable from
http://www.zib.de/groetschel/pubnew/paper/groetschelpadberg2001a.pdf
Martin
Grötschel
16

Ulysses

The distance table

Martin
Grötschel
17

Ulysses roundtrip

optimal „Ulysses tour“
Martin
Grötschel
18

Malen nach Zahlen
TSP in art ?

 When was this invented?
Martin
Grötschel
19

Survey Books
Literature: more than 1000 entries in Zentralblatt/Math

Zbl 0562.00014 Lawler, E.L.(ed.); Lenstra, J.K.(ed.); Rinnooy
Kan, A.H.G.(ed.); Shmoys, D.B.(ed.)
The traveling salesman problem. A guided tour of
combinatorial optimization. Wiley-Interscience Series in Discrete
Mathematics. A Wiley-Interscience publication. Chichester etc.: John
Wiley \& Sons. X, 465 p. (1985). MSC 2000: *00Bxx 90-06

Zbl 0996.00026 Gutin, Gregory (ed.); Punnen, Abraham P.(ed.)
The traveling salesman problem and its variations.
Combinatorial Optimization. 12. Dordrecht: Kluwer Academic
Publishers. xviii, 830 p. (2002). MSC 2000: *00B15 90-06 90Cxx
Martin
Grötschel
20

Contents
1. Introduction
2. The TSP and some of its history
3. The TSP and some of its variants
4. Some applications
5. Modeling issues
6. Heuristics
7. How combinatorial optimizers do it

Martin
Grötschel
The Travelling Salesman Problem
21

and Some of its Variants
 The symmetric TSP
 The asymmetric TSP
 The TSP with precedences or time windows
 The online TSP
 The symmetric and asymmetric m-TSP
 The price collecting TSP
 The Chinese postman problem
(undirected, directed, mixed)
 Bus, truck, vehicle routing
 Edge/arc & node routing with capacities
 Combinations of these and more

Martin
Grötschel
22

http://www.densis.fee.unicamp.br/~
moscato/TSPBIB_home.html

Martin
Grötschel
23

Contents
1. Introduction
2. The TSP and some of its history
3. The TSP and some of its variants
4. Some applications
5. Modeling issues
6. Heuristics
7. How combinatorial optimizers do it

Martin
Grötschel
24

Production of ICs and PCBs

Integrated Circuit (IC)       Printed Circuit Board (PCB)

Problems: Logical Design, Physical Design
Correctness, Simulation, Placement of
Components, Routing, Drilling,...
Martin
Grötschel
25

Correct modelling of a
printed circuit board drilling problem
length of a
move of the
drilling head:
Euclidean norm,
Max norm,
Manhatten norm?

2103 holes to be drilled
Martin
Grötschel
26

Drilling 2103 holes into a PCB

Significant Improvements
via TSP

(due to Padberg & Rinaldi)

Martin
Grötschel
industry solution       optimal solution
Siemens-Problem
PCB da4
Martin Grötschel, Michael Jünger, Gerhard Reinelt,
Optimal Control of Plotting and Drilling Machines:
A Case Study, Zeitschrift für Operations Research,
35:1 (1991) 61-84
http://www.zib.de/groetschel/pubnew/paper/groetscheljuengerreinelt1991.pdf

before                                                  after
Siemens-Problem
PCB da1

Grötschel, Jünger, Reinelt

before                      after
29

Martin
Grötschel
30

Leiterplatten-Bohrmaschine
Printed Circuit Board Drilling Machine

Martin
Grötschel
31

Foto einer Flachbaugruppe
(Leiterplatte)

Martin
Grötschel
32

Foto einer Flachbaugruppe
(Leiterplatte) - Rückseite

Martin
Grötschel
33

442 holes to be drilled

Martin
Grötschel
34

Typical PCB drilling problems at
Siemens

da1             da2       da3       da4

Number of holes       2457             423      2203      2104
Number of drills         7               7         6        10
Tour length        3518728         1049956   1958161   4347902

Table 4

Martin
Grötschel
35

Fast heuristics

da1          da2       da3       da4

CPU time (min:sec)       1:58         0:05      1:43      1:43
Tour length           1695042       984636   1642027   1928371

Improvement in %        56.87        14.60     26.94     58.38

Table 5

Martin
Grötschel
36

Optimizing the stacker cranes of a
Siemens-Nixdorf warehouse

Martin
Grötschel
37

Herlitz at Falkensee (Berlin)

Martin
Grötschel
Example: Control of the stacker
38

cranes in a Herlitz warehouse

Martin
Grötschel
39

Logistics of collecting
electronics garbage

Andrea Grötschel
Diplomarbeit (2004)

Martin
Grötschel
40

Location plus tour planning (m-TSP)

Martin
Grötschel
41

The Dispatching Problem at ADAC:
an online m-TSP

Dispatching Center (Pannenzentrale)

Data Transm.

Dispatcher
„Gelber Engel“
Martin
Grötschel
Online-TSP (in a metric space)
Instance:   r1 , r2 ,   , rn where ri  (ti , xi )
x1                              x1       x2

0                               0
t  t1                        t  t2

Goal:     Find fastest tour serving all requests
(starting and ending in 0)
Algorithm ALG is c-competitive if
ALG    c  OPT  
for all request sequences 
43

Implementation competitions

Martin
Grötschel
44

Contents
1. Introduction
2. The TSP and some of its history
3. The TSP and some of its variants
4. Some applications
5. Modeling issues
6. Heuristics
7. How combinatorial optimizers do it

Martin
Grötschel
45

LP Cutting Plane Approach

Even MODELLING is not easy!

What is the „right“ LP relaxation?

N. Ascheuer, M. Fischetti, M. Grötschel,
„Solving the Asymmetric Travelling Salesman
Problem with time windows by branch-and-cut“,
Mathematical Programming A (2001), see
http://www.zib.de/groetschel/pubnew/paper/ascheuerfischettigroetschel2001.pdf

Martin
Grötschel
46

IP formulation of the asymmetric TSP

min cT x
x( (i ))

1    i  V  0
x(  (i ))     1    i  V  0
x( A(W )) | W | 1   W  V  0 , 2 | W | n
xij  0,1           (i, j )  A.

Martin
Grötschel
47

Time Windows
 This is a typical situation in delivery problems.
 Customers must be served during a certain
period of time, usually a time interval is given.
 access to pedestrian areas
 opening hours of a customer
 delivery to assembly lines
 just in time processes

Martin
Grötschel
48

Model 1
T
min c x
x(  (i))                  1      i  V  0
x(  (i))                  1      i  V  0
ti  ij  (1  xij )  M    tj    i, j  A, j  0
ri        ti                di    i  V
ti                          N      i  V  0
xij                          0,1 i, j  A.
Martin
Grötschel
49

Model 2
min cT x
x(  (i ))      1      i  V  0
x(  (i ))      1      i  V  0
x( A(W ))  | W | 1     W  V  0 , 2 | W | n
x( P )  | P | 1  k  2 infeasible path P  (v1 , v2 ,   , vk )
xij  0,1              (i, j )  A.

Martin
Grötschel
50

Model 3
T
min c x
x(  (i ))                             1          i  V  0
x(  (i ))                             1          i  V  0
n               n                         n

 y  
i 1
ij
i 0
ij    xij         y jk j  V
k 1
i j            i j                      k j

ri  xij  yij  di  xij               i, j  0,     , n, i  j , i  0
xij                            0,1               (i, j )  A
yij                            0,1, 2,...        (i, j )  A
Martin
Grötschel
Model 1, 2, 3
min cT x                                               min cT x
x(  (i))                  1      i  V  0       x(  (i ))                            1          i  V  0
x(  (i))                  1      i  V  0       x(  (i ))                            1          i  V  0
ti  ij  (1  xij )  M    tj    i, j  A, j  0    n              n                         n

ri       ti                 di    i  V              y  
i 1
ij
i 0
ij    xij         y jk j  V
k 1
i  V  0
i j           i j                      k j
ti                          N
ri  xij  yij  di  xij              i, j  0,     , n, i  j , i  0
xij                          0,1 i, j  A.
xij                           0,1               (i, j )  A
yij                           0,1, 2,...        (i, j )  A

min cT x
x(  (i ))           1     i  V  0
x(  (i ))           1     i  V  0
x( A(W )) | W | 1          W  V  0 , 2 | W | n
x( P ) | P | 1  k  2     infeasible path P  (v1 , v2 ,        , vk )
xij  0,1                  (i, j )  A.
52

Cutting Planes Used for all Three Models
(Separation Routines)
    Subtour Elimination Constraints (SEC)
    2-Matching Constraints
    , ,( , )-Inequalities
    "Special“ Inequalities and PCB-Inequalities
    Dk-Inequalities
    Infeasible Path Elimination Constraints (IPEC)
    Strengthened  ,  -Inequalities
    Two-Job Cuts
    Pool Separation
    SD-Inequalities
    + various strengthenings/liftings

Martin
Grötschel
53

Further Implementation Details
 Preprocessing
   Tightening Time Windows
   Release and Due Date Adjustment
   Construction of Precedences
   Elimination of Arcs
    Branching (only on x-variables)
    Enumeration Strategy (DFS, Best-FS)
    Pricing Frequency (every 5th iteration)
    Tailing Off
    LP-exploitation Heuristics (after a new feasible LP
solution is found),
they outperform the other heuristics
Martin
Grötschel
54

Results
 Very uneven performance
 Model 1 is really bad in general
 Model 2 is best on the average (winner in 16
of 22 test cases)
 Model 3 is better when few time windows are
active (6 times winner, last in all other cases,
severe numerical problems, very difficult LPs)

How could you have guessed?
Martin
Grötschel
Unevenness of Computational
55

Results

problem   #nodes      gap    #cutting     #LPs      time
planes
rbg041a        43   9.16%    > 1 mio    109,402 > 5 h

rbg067a        69      0%        176          2     6 sec

Largest problem solved to optimality: 127 nodes
Largest problem not solved optimally: 43 nodes

Martin
Grötschel
56

Contents
1. Introduction
2. The TSP and some of its history
3. The TSP and some of its variants
4. Some applications
5. Modeling issues
6. Heuristics
7. How combinatorial optimizers do it

Martin
Grötschel
57

Need for Heuristics
 Many real-world instances of hard combinatorial
optimization problems are (still) too large for exact
algorithms.
 Or the time limit stipulated by the customer for the
solution is too small.
 Therefore, we need heuristics!
 Exact algorithms usually also employ heuristics.
 What is urgently needed is a decision guide:

Which heuristic will most likely work well on what
problem ?
Martin
Grötschel
58

Primal and Dual Heuristics
 Primal Heuristic: Finds a (hopefully) good feasible solution.
 Dual Heuristic: Finds a bound on the optimum solution value
(e.g., by finding a feasible solution of the LP-dual of an LP-
relaxation of a combinatorial optimization problem).

Minimization:

dual heuristic value ≤ optimum value ≤ primal heuristic value

quality guarantee
in practice and theory
Martin
Grötschel
59

Heuristics: A Survey
 Greedy Algorithms
 Exchange & Insertion Algorithms
 Neighborhood/Local Search
 Variable Neighborhood Search, Iterated Local Search
 Random sampling
 Simulated Annealing
 Taboo search
 Great Deluge Algorithms
 Simulated Tunneling
 Neural Networks
 Scatter Search
 Greedy Randomized Adaptive Search Procedures
Martin
Grötschel
60

Heuristics: A Survey
 Genetic, Evolutionary, and similar Methods
 DNA-Technology
 Ant and Swarm Systems
 (Multi-) Agents
 Population Heuristics
 Memetic Algorithms (Meme are the “missing links” gens and
mind)
 Fuzzy Genetics-Based Machine Learning
 Fast and Frugal Method (Psychology)
 Method of Devine Intuition (Psychologist Thorndike)

 …..
Martin
Grötschel
61

Heuristics: A Survey
Currently best heuristic with respect to worst-case guarantee:
Christofides heuristic
 compute shortest spanning tree
 compute minimum perfect 1-matching of graph induced by the odd
nodes of the minimum spanning tree
 the union of these edge sets is a connected Eulerian graph
 turn this graph into a tour by making short-cuts.
For distance functions satisfying the triangle inequality, the resulting tour
is at most 50% above the optimum value

Martin
Grötschel
62

Understanding Heuristics,
Approximation Algorithms
 worst case analysis
 There is no polynomial time approx. algorithm for STSP/ATSP.
 Christofides algorithm for the STSP with triangle inequality
 average case analysis
 Karp‘s analysis of the patching algorithm for the ATSP
 probabilistic problem analysis
 for Euclidean STSP in unit square, TSP constant 1.714.. n
 polynomial time approximation schemes (PAS)
 Arora‘s polynomial-time approximation schemes for
Euclidean STSPs
 fully-polynomial time approximation schemes (FPAS)
 not for TSP/ATSP but, e.g., for knapsack (Ibarra&Kim)

 These concepts – unfortunately – often do not really help to guide
practice.
 experimental evaluation
 Lin-Kernighan for STSP (DIMACS challenges))
Martin
Grötschel
63

Contents
1. Introduction
2. The TSP and some of its history
3. The TSP and some of its variants
4. Some applications
5. Modeling issues
6. Heuristics
7. How combinatorial optimizers do it

Martin
Grötschel
64

Polyhedral Theory (of the TSP)
STSP-, ATSP-,TSP-with-side-constraints-
Polytope:= Convex hull of all incidence
vectors of feasible tours
To be investigated:
 Dimension
 Equation system defining the affine hull
 Facets
 Separation algorithms
Martin
Grötschel
65

The symmetric travelling salesman polytope

QTn : conv{ T  Z E | T tour in K n }   (  ij  1 if ij  T , else  0)
T

 {x  R E | x( (i ))  2         i  V
x( E (W )) | W | 1 W  V \ 1 ,3 | W | n  3
0  xij  1           ij  E}

min cT x
x( (i ))  2        i  V
x( E (W )) | W | 1 W  V \ 1 ,3 | W | n  3
xij  0,1           ij  E

 The LP relaxation is solvable in polynomial time
Martin
Grötschel
66

Relation between IP and LP-relaxation

Open Problem:
 If costs satisfy the triangle inequality, then
IP-OPT <= 4/3 LP-SEC
IP-OPT <= 3/2 LP-SEC (Wolsey)

Martin
Grötschel
67

General cutting plane theory:
Gomory Mixed-Integer Cut
 Given y, x j  ¢  and
,
y   aij x j  d  d   f , f  0
 
 Rounding: Where aij  aij   f j , define
 
                         
t  y    aij  x j : f j  f    aij  x j : f j  f  ¢
                                                        
 Then
 f x j    j   : f j  f     f j  1x j : f j  f  d  t
 Disjunction:
t  d     f j x j : f j  f   f
 
                  
t   d    1  f j  x j : f j  f  1  f   
 Combining

  f   j                                                                
f  x j : f j  f   1  f j  1  f   x j : f j  f  1
                    
Martin
Grötschel
68

clique trees
 A clique tree is a connected graph C=(V,E), composed of
cliques satisfying the following properties

Martin
Grötschel
69

Polyhedral Theory of the TSP
Comb inequality

2-matching
constraint

handle
tooth
Martin
Grötschel
70

Clique Tree Inequalities

Martin
Grötschel
71

Clique Tree Inequalities
http://www.zib.de/groetschel/pubnew/paper/groetschelpulleyblank1986.pdf

h                t                h

 x (  ( H ))   x (  (T ))   | H
i               j                i
|  h  2t
i 1             j 1             i 1

h                t                    h                 t
t 1
 x ( E ( H ))   x ( E (T ))   | H
i                  j                i
|    (| T   j
| t j ) 
i 1             j 1              i 1                 i 1                       2

Hi, i=1,…,h are the handles
Tj, j=1,…,t are the teeth
tj is the number of handles
that tooth Tj intersects

Martin
Grötschel
72

Valid Inequalities for STSP
   Trivial inequalities
   Degree constraints
   Subtour elimination constraints
   2-matching constraints, comb inequalities
   Clique tree inequalities (comb)
   Bipartition inequalities (clique tree)
   Path inequalities (comb)
   Star inequalities (path)
   Binested Inequalities (star, clique tree)
   Ladder inequalities (2 handles, even # of teeth)
   Domino inequalities
   Hypohamiltonian, hypotraceable inequalities
Martin
 etc.
Grötschel
73

A very special case

Petersen graph, G = (V, F),
the smallest hypohamiltonian graph

x( F )  9 defines a facet of QT
10

but not a facet of QT , n  11
n

M. Grötschel & Y. Wakabayashi

Martin
Grötschel
74

Hypotraceable graphs and the STSP
On the right is the smallest
known hypotraceable graph
(Thomassen graph, 34 nodes).
Such graphs have no
hamiltonian path, but when
any node is deleted, the
remaining graph has a
hamiltonian path.
How do such graphs induce
inequalities valid for the
symmetric travelling salesman
polytope?

For further information see:
Martin
Grötschel
http://www.zib.de/groetschel/pubnew/paper/groetschel1980b.pdf
“Wild facets of the asymmetric
75

travelling salesman polytope”
 Hypohamiltonian and hypotraceable directed graphs also exist and
induce facets of the polytopes associated with the asymmetric TSP.

 Information “hypohamiltonian” and “hypotraceable” inequalities can
be found in
http://www.zib.de/groetschel/pubnew/paper/groetschelwakabayashi1981a.pdf
http://www.zib.de/groetschel/pubnew/paper/groetschelwakabayashi1981b.pdf

Martin
Grötschel
76

Valid and facet defining inequalities for
STSP: Survey articles

 M. Grötschel, M. W. Padberg (1985 a, b)

 M. Jünger, G. Reinelt, G. Rinaldi (1995)

 D. Naddef (2002)

 The TSP book (ABCC, 2006)

Martin
Grötschel
77

Counting Tours and Facets
n    # tours     # different facets   # facet classes
3         1                      0                 0
4         3                      3                 1
5        12                     20                 2
6        60                    100                 4
7       360                  3,437                 6
8      2520               194,187                 24
9     20,160           42,104,442                192
10   181,440   >= 52,043,900,866          >=15,379
Martin
Grötschel
78

Separation Algorithms
 Given a system of valid inequalities (possibly
of exponential size).
 Is there a polynomial time algorithm (or a
good heuristic) that,
 given a point,
 checks whether the point satisfies all inequalities
of the system, and
 if not, finds an inequality violated by the given
point?

Martin
Grötschel
79

Separation

K

Grötschel, Lovász, Schrijver (GLS):
“Separation and optimization
are polynomial time equivalent.”
Martin
Grötschel
80

Separation Algorithms
 There has been great success in finding exact
polynomial time separation algorithms, e.g.,
 for subtour-elimination constraints
 for 2-matching constraints (Padberg&Rao, 1982)

 or fast heuristic separation algorithms, e.g.,
 for comb constraints
 for clique tree inequalities

 and these algorithms are practically efficient

Martin
Grötschel
81

Polyhedral Combinatorics
 This line of research has resulted in
powerful cutting plane algorithms for
combinatorial optimization problems.
 They are used in practice to solve
exactly or approximately (including
branch & bound) large-scale real-world
instances.

Martin
Grötschel
82

Deutschland
15,112
D. Applegate, R.Bixby,
V. Chvatal, W. Cook
15,112
cities
114,178,716
variables

2001

Martin
Grötschel
83

How do we solve a TSP like this?

 Upper bound:               Lower bound:
Heuristic search             Linear programming

 Chained Lin-Kernighan      Divide-and-conquer
 Polyhedral combinatorics
 Parallel computation
 Algorithms & data structures

The LOWER BOUND is the mathematically and
algorithmically hard part of the work
Martin
Grötschel
84

Work on LP relaxations of the
symmetric travelling salesman polytope

QTn : conv{ T  Z E | T tour in K n }

T
min c x
x( (i ))  2       i  V
x( E (W )) | W | 1 W  V \ 1 ,3 | W | n  3
0  xij  1          ij  E
xij  0,1          ij  E
 Integer Programming Approach
Martin
Grötschel
85

cutting plane technique for integer and
mixed-integer programming

Feasible
integer
solutions
Objective
function
Convex
hull
LP-based
relaxation
Cutting
planes
Martin
Grötschel
86

Clique-tree cut for pcb442 from B. Cook

Martin
Grötschel
87

LP-based Branch & Bound
Solve LP relaxation:
Root     v=0.5 (fractional)     Upper Bound

G
A
P
Integer                                         Lower Bound

Infeas

Integer

Remark: GAP = 0  Proof of optimality

Martin
Grötschel
88

A Branching
Tree
Applegate
Bixby
Chvátal
Cook

Martin
Grötschel
89

Managing the LPs of the TSP

|V|(|V|-1)/2

CORE LP              Column generation: Pricing.

~ 3|V| variables
astronomical

Cuts: Separation

~1.5|V| constraints

Martin
Grötschel
90

A Pictorial History of Some
TSP World Records

Martin
Grötschel
Some TSP World Records
91

2006            year     authors       # cities     # variables
pla 85,900
solved          1954           DFJ       42/49       820/1,146
3,646,412,050        1977              G        120            7,140
variables
1987            PR         532         141,246
number of cities     1988            GH         666         221,445
2000x
increase          1991            PR       2,392        2,859,636

4,000,000         1992         ABCC        3,038        4,613,203
times          1994         ABCC         7,397      27,354,106
problem size
increase         1998         ABCC       13,509       91,239,786

in 52         2001         ABCC       15,112      114,178,716
years         2004         ABCC       24,978      311,937,753
Martin  2005 W. Cook, D. Epsinoza, M. Goycoolea
Grötschel
33,810     571,541,145
92

The current champions
ABCC stands for
D. Applegate, B. Bixby, W. Cook, V. Chvátal

 almost 15 years of code development
 presentation at ICM’98 in Berlin, see proceedings
 have made their code CONCORDE available in
the Internet

Martin
Grötschel
93

USA 49

49 cities
1,146 variables

1954

G. Dantzig, D.R. Fulkerson, S. Johnson
Martin
Grötschel
94

West-Deutschland und Berlin

120 Städte
7140 Variable

1975/1977/1980

M. Grötschel

Martin
Grötschel
95

A tour around the world
666 cities
221,445 variables

1987/1991

M. Grötschel, O. Holland, see
Martin
Grötschel
http://www.zib.de/groetschel/pubnew/paper/groetschelholland1991.pdf
96

USA cities with population >500

13,509
cities
91,239,786
Variables

1998

D. Applegate, R.Bixby, V. Chvátal, W. Cook
Martin
Grötschel
97

usa13509: The branching tree

0.01%
initial gap

Martin
Grötschel
98

Summary: usa13509
 9539 nodes branching tree
 48 workstations (Digital Alphas, Intel
Pentium IIs, Pentium Pros, Sun
UntraSparcs)

 Total CPU time:   4 cpu years

Martin
Grötschel
99

Overlay of
3 Optimal
Germany
tours

from
ABCC 2001

http://www.math.princeton.edu/
tsp/d15sol/dhistory.html

Martin
Grötschel
100

Optimal Tour of Sweden

311,937,753
variables

ABCC
plus
Keld Helsgaun
Roskilde Univ.
Denmark.

Martin
Grötschel
101

World Tour, current status
http://www.tsp.gatech.edu/world/

We give links to several images of the World TSP tour
of length 7,516,353,779 found by Keld Helsgaun in
December 2003. A lower bound provided by the
Concorde TSP code shows that this tour is at most
0.076% longer than an optimal tour through the
Martin      1,904,711 cities.
Grötschel
The Travelling Salesman Problem
a brief survey

Martin Grötschel
Summary of Chapter 2
The END
of the class
Polyhedral Combinatorics (ADM III)
May 18, 2010

Martin Grötschel     Institute of Mathematics, Technische Universität Berlin (TUB)
 DFG-Research Center “Mathematics for key technologies” (MATHEON)
 Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)
groetschel@zib.de   http://www.zib.de/groetschel

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