The Travelling Salesman Problem: A brief survey Martin Grötschel

Vorausschau auf die Vorlesung Das Travelling-Salesman-Problem (ADM III) im WS 2013/14

Martin Grötschel [email protected]

14. Oktober 2013

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

2

Contents

1.

2.

3.

4.

5.

6.

7.

Introduction The TSP and some of its history The TSP and some of its variants Some applications Modeling issues Heuristics How combinatorial optimizers do it

Martin Grötschel

3

Contents

1.

2.

3.

4.

5.

6.

7.

Introduction The TSP and some of its history The TSP and some of its variants Some applications Modeling issues Heuristics How combinatorial optimizers do it

Martin Grötschel

4 Martin Grötschel

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)  

e

 I 

I

,

where I

 2

E and E finite

An important issue: How is I given?

5 Martin Grötschel Special „simple“ combinatorial optimization problems

Finding a   minimum spanning tree in a graph shortest path in a directed graph   maximum matching in a graph 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.

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 Martin Grötschel

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 asymmetric ( ATSP ).

http://www.tsp.gatech.edu/

9 THE TSP book

suggested reading for everyone interested in the TSP

Martin Grötschel

10

Another recommendation Bill Cook‘s new book

Martin Grötschel

11 Martin Grötschel

The travelling salesman problem

Two mathematical formulations of the TSP 1.

Version

:

Let K n



and let c be the length of e e

 )

T n

.

 H}.

Find

H

be the set of all

2.

Version

:

Find a cyclic permutation



of i n

  1

c i



is as small as possible

.

 Does that help solve the TSP?

12

Contents

1.

2.

3.

4.

5.

6.

7.

Introduction The TSP and some of its history The TSP and some of its variants Some applications Modeling issues Heuristics How combinatorial optimizers do it

Martin Grötschel

13 Martin Grötschel

Usually quoted as the forerunner of the TSP Usually quoted as the origin of the TSP

about 100 years earlier

15 Martin Grötschel

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

16

A TSP contest 1962: 10.000 $ Prize

Martin Grötschel

17 Ulysses roundtrip (an even older TSP ?) Martin Grötschel

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

18

Ulysses

Martin Grötschel

The distance table

19

Ulysses roundtrip

Martin Grötschel

optimal „Ulysses tour“

20 Malen nach Zahlen TSP in art ?

Martin Grötschel

 When was this invented?

21

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

Martin Grötschel

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

22

Contents

1.

2.

3.

4.

5.

6.

7.

Introduction The TSP and some of its history The TSP and some of its variants Some applications Modeling issues Heuristics How combinatorial optimizers do it

Martin Grötschel

23

The Travelling Salesman Problem 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

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

Martin Grötschel

25

Contents

1.

2.

3.

4.

5.

6.

7.

Introduction The TSP and some of its history The TSP and some of its variants Some applications Modeling issues Heuristics How combinatorial optimizers do it

Martin Grötschel

26 Production of ICs and PCBs Martin Grötschel

Integrated Circuit (IC) Printed Circuit Board (PCB) Problems: Logical Design, Physical Design Correctness, Simulation, Placement of Components, Routing, Drilling ,...

27 Correct modelling of a printed circuit board drilling problem

length of a move of the drilling head : Euclidean norm, Max norm, Manhatten norm?

Martin Grötschel

2103 holes to be drilled

28

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

31 Martin Grötschel

32

Leiterplatten-Bohrmaschine

Printed Circuit Board Drilling Machine Martin Grötschel

33 Foto einer Flachbaugruppe (Leiterplatte) Martin Grötschel

34 Foto einer Flachbaugruppe (Leiterplatte) - Rückseite Martin Grötschel

35

442 holes to be drilled

Martin Grötschel

36 Typical PCB drilling problems at Siemens

Number of holes Number of drills Tour length da1 da2 da3 da4 2457 7 3518728 423 7 1049956 2203 6 1958161 2104 10 4347902 Table 4

Martin Grötschel

37

CPU time (min:sec) Tour length Improvement in %

Fast heuristics

da1 da2 da3 da4 1:58 1695042 56.87

Table 5 0:05 984636 1:43 1642027 1:43 1928371 14.60

26.94

58.38

Martin Grötschel

38 Optimizing the stacker cranes of a Siemens-Nixdorf warehouse Martin Grötschel

39

Herlitz at Falkensee (Berlin)

Martin Grötschel

40

Example: Control of the stacker cranes in a Herlitz warehouse

Martin Grötschel

41 Logistics of collecting electronics garbage

Andrea Grötschel Diplomarbeit (2004)

Martin Grötschel

42 Location plus tour planning (m-TSP) Martin Grötschel

43 The Dispatching Problem at ADAC: an online m-TSP

„Gelber Engel“ Dispatching Center (Pannenzentrale) Data Transm.

Dispatcher

Martin Grötschel

Online-TSP (in a metric space)

Instance:   1 , , 2

x

1 ,

r n

where

r i

 ( , )

i i x

1

x

2 0

t



t

2 0

t



t

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

45

Implementation competitions

Martin Grötschel

46

Contents

1.

2.

3.

4.

5.

6.

7.

Introduction The TSP and some of its history The TSP and some of its variants Some applications Modeling issues Heuristics How combinatorial optimizers do it

Martin Grötschel

47

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

48

IP formulation of the asymmetric TSP min

T c x x

(  

x

(  

W

 1  1

x ij

 

V

      

A

.

W

| 

n

Martin Grötschel

49

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

50 Martin Grötschel

Model 1

min

T c x x

(  

x

(  

t i r i

 

ij



t i t i x ij x ij

) 

M

 1  1 

t j



d i



N

        0  

A

.

51

Model 2

min

T c x x

(  

x

(   ))  |

W

 |

P

 1  1

k x ij

 2

V

     

W

| 

n

 infeasible path

P

 ( , 1 2 ,  

A

.

,

v k

)

Martin Grötschel

52 Martin Grötschel

Model 3

min

T c x x

(  

x

(    1  1

V V i n

  1 

y ij i



ij

 

i n

  0 

ij



x ij y ij

 

i ij x ij y ij



k k n

  1 

j y jk

   0,1, 2,...

  0, 

V

 

A

 ( , ) 

A

     0

min

x

(  

x

(  

t i r i

 

ij



t i x ij t i x ij

) 

M

 1  1 

t j



d i



N

  

Model 1, 2, 3

      0 

A

.

min

T c x x

(  

x

(    1  1

V V i n

  1 

y ij i



ij

 

i n

  0 

ij



x ij y ij

 

i ij x ij y ij



k k n

  1 

j

   0,1, 2,...



y jk

 0, 

V

 

A

 ( , ) 

A

     0 min

x

(  

x

(  

x ij

 

P W

 1  1

k

2 

W V V

       infeasible path

P W

| 

n

 ( , 1 2 ,  

A

.

,

v k

)

54 Martin Grötschel

          

Cutting Planes Used for all Three ( Separation Routines) Models

Subtour Elimination Constraints (SEC) 2-Matching Constraints     -Inequalities "Special“ Inequalities and PCB-Inequalities D

k

-Inequalities Infeasible Path Elimination Constraints (IPEC) Two-Job Cuts Pool Separation SD-Inequalities + various strengthenings/liftings

55 Martin Grötschel

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

56 Martin Grötschel

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?

57

Unevenness of Computational Results

problem #nodes rbg041a gap 43 9.16% #cutting planes > 1 mio #LPs time 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

58

Contents

1.

2.

3.

4.

5.

6.

7.

Introduction The TSP and some of its history The TSP and some of its variants Some applications Modeling issues Heuristics How combinatorial optimizers do it

Martin Grötschel

59 Martin Grötschel

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 ?

60

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).

Martin Grötschel

Minimization: dual heuristic value ≤ optimum value ≤ primal heuristic value

quality guarantee

in practice and theory

61 Martin Grötschel

           

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

62

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

 …..

63

The typical heuristics junk

 Hyper-heuristics in Co-operative Search The interest in parallel co-operative approaches has risen considerably due to, not only the availability of co-operative environments at low cost, but also their success to provide novel ways to combine different (meta-)heuristics. Current research has shown that the parallel execution and co-operation of several (meta-)heuristics could improve the quality of the solutions that each of them would be able to find by itself working on a standalone basis. Moreover, parallel and distributed approaches can be used to provide more powerful and robust problem solving environments in a variety of problem domains. Hyper-heuristics, on the other hand, represent a set of search methodologies which are applicable to different problem domains. They aim to raise the level of generality, for example by choosing and/or generating new methodologies on demand during the search process. The goal of this study is to explore the cooperative search mechanisms within a hyper-heuristic framework. This exciting research area lies at the interface between operational research and computer science and involves understanding of (distributed) decision making mechanisms and learning, design, implementation and analysis of automated search methodologies. The application domains will be cross disciplinary.

Martin Grötschel

64

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

65

    

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..

polynomial time approximation schemes (PAS)

n

 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)

Martin Grötschel

  These concepts – unfortunately – often do not really help to guide practice.

experimental evaluation  Lin-Kernighan for STSP (DIMACS challenges))

66

Contents

1.

2.

3.

4.

5.

6.

7.

Introduction The TSP and some of its history The TSP and some of its variants Some applications Modeling issues Heuristics How combinatorial optimizers do it

Martin Grötschel

67 Martin Grötschel

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

68 The symmetric travelling salesman polytope

Q T n

: 

conv

{ 

T

 {

x



R

E



Z

E

|

T tour in K n

}   2

W

0 

x ij

 1

W

( 

ij T

 1

if ij



V E

}  

W

|

n

3  0)

Martin Grötschel

min

T c x x

  2

W x ij



W E V

 

W

|

n

3  The LP relaxation is solvable in polynomial time

69 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

70 Martin Grötschel General cutting plane theory: Gomory Mixed-Integer Cut

     ,

j

 ў  ,

y

 

a x ij j



f

,

f

 0 Rounding : Where define

ij a ij j t

Then

t

  

f x j j

 : Disjunction : 

t

 Combining  

f j

 

a ij

   

f

1

x



f x j

 

j j

:

f

:

f



j j f j

  

x j f



j f

:

f



f j

   1 

x

 

j f f

:

f



j a ij x f j

:

d f j



t f

  ў   

f j j

:

f j



f

     1 

f j

  1 

f



x j

:

f j



f

  1

71

clique trees

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

Martin Grötschel

72

Polyhedral Theory of the TSP

Comb inequality 2-matching constraint

Martin Grötschel

handle tooth

73

Clique Tree Inequalities

Martin Grötschel

74 Martin Grötschel

i

Clique Tree Inequalities

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

i h

  1

x H i

)) 

j t

  1

x



T j

)) 

i h

  1 |

H i

| 2

t h

  1

x H i

)) 

j t

  1

T j

)) 

h



i

 1 |

H i

| 

t



i

 1 (|

T j

| 

t j

) 

t

 1 2 H i , i=1,…,h are the handles T j , j=1,…,t are the teeth t j is the number of handles that tooth T j intersects

75

Valid Inequalities for STSP

Martin Grötschel

             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 etc.

76

A very special case

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

Martin Grötschel

 9

defines a facet of Q

10

T n T

M. Grötschel & Y. Wakabayashi  11

77 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?

Martin Grötschel

For further information see: http://www.zib.de/groetschel/pubnew/paper/groetschel1980b.pdf

78

“Wild facets of the asymmetric travelling salesman polytope”

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

Martin Grötschel

 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

79 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

80 Martin Grötschel

n 3 4 5 6 7 8 9 10

Counting Tours and Facets

# tours 1 3 12 60 360 2520 20,160 181,440 # different facets 0 3 20 100 3,437 194,187 42,104,442 >= 52,043,900,866 # facet classes 0 1 2 24 192 4 6 >=15,379

81

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

82

Separation

K

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

Martin Grötschel

83

  

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

84

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

85 Deutschland 15,112

D. Applegate, R.Bixby, V. Chvatal, W. Cook 15,112 cities 114,178,716 variables 2001

Martin Grötschel

86 How do we solve a TSP like this?



Upper bound:

Heuristic search  Chained Lin-Kernighan 

Lower bound:

     Linear programming Divide-and-conquer Polyhedral combinatorics Parallel computation Algorithms & data structures

Martin Grötschel

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

87 Work on LP relaxations of the symmetric travelling salesman polytope

Q T n

: 

conv

{ 

T



Z

E

|

T tour in K n

}

Martin Grötschel

min

T c x x

  2

W W V

  0 

x ij x ij

  1

E E

 Integer Programming Approach

W

|

n

3

88 cutting plane technique for integer and mixed-integer programming Martin Grötschel

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

89

Clique-tree cut for pcb442

from B. Cook

Martin Grötschel

90

LP-based Branch & Bound

Solve LP relaxation: v=0.5 (fractional) Root Upper Bound G A P Lower Bound Integer Infeas Integer Remark: GAP = 0  Proof of optimality

Martin Grötschel

91 A Branching Tree

Applegate Bixby Chvátal Cook

Martin Grötschel

92

Managing the LPs of the TSP

CORE LP

|

V

|(|

V

|-1)/2 Column generation: Pricing.

~ 3|V| variables ~1.5|V| constraints Martin Grötschel

93 A Pictorial History of Some TSP World Records Martin Grötschel

94 Some TSP World Records

2006 pla 85,900 solved 3,646,412,050 variables year 1954 1977 authors DFJ G # cities 42/49 120 1987 PR 532 number of cities 2000x increase 1988 1991 GH PR 666 2,392 4,000,000 times problem size increase in 52 years 1992 1994 1998 2001 2004 ABCC ABCC ABCC ABCC ABCC 3,038 7,397 13,509 15,112 24,978 # variables 820/1,146 7,140 141,246 221,445 2,859,636 4,613,203 27,354,106 91,239,786 114,178,716 311,937,753

Martin Grötschel

2005 W. Cook, D. Epsinoza, M. Goycoolea 33,810 571,541,145

95

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

96

USA 49

Martin Grötschel

G. Dantzig, D.R. Fulkerson, S. Johnson 49 cities 1,146 variables 1954

97

West-Deutschland und Berlin

120 Städte 7140 Variable 1975/1977/1980 M. Grötschel

Martin Grötschel

98

A tour around the world

666 cities 221,445 variables 1987/1991

Martin Grötschel

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

99

USA cities with population >500

13,509 cities 91,239,786 Variables 1998

Martin Grötschel

D. Applegate, R.Bixby, V. Chvátal, W. Cook

100

usa13509: The branching tree

0.01% initial gap

Martin Grötschel

101

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

102 Overlay of 3 Optimal Germany tours

from ABCC 2001 http://www.math.princeton.edu/ tsp/d15sol/dhistory.html

Martin Grötschel

103 Optimal Tour of Sweden

311,937,753 variables ABCC plus Keld Helsgaun Roskilde Univ. Denmark.

Martin Grötschel

104

World Tour, current status

http://www.tsp.gatech.edu/world/

Martin Grötschel

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 1,904,711 cities.

105 Martin Grötschel

      

Vorlesungsplan

Kapitel 1. Das Travelling-Salesman- und verwandte Probleme: ein Überblick und Anwendungen 1. Vorlesung: ppt-Überblick über das TSP, alte Folien und Cook Book, Archäologie, Dantzig, Fulkerson und Johnson Kapitel 2. Hamiltonsche und hypohamiltonsche Graphen und Digraphen 2. Vorlesung: Hamiltonsche Graphen aus Bondy und Murty 3. Vorlesung: Hypohamiltonsche und Hypobegehbare Graphen (Thomassen-Paper und Paper mit Yoshiko) Kapitel 3. Die „natürlichen“ IP-Formulierungen des TSP und des ATSP, Travelling Salesman-Polytope Subtour-Formulierungen, STSP und ATSP-Polytop Kapitel 4. Kombinatorische Verwandte des TSP Das 1-Baum-, 2-Matching-, Zuordnungs-, 1-Arboreszenz-Problem

106 Martin Grötschel

      

Vorlesungsplan

Kapitel 5. Gütegarantien für Heuristiken, Eröffnungsheuristiken für das TSP (NN, Insert, Christofides,...) Kapitel 6. Verbesserungsheuristiken und ein polynomiales Approximationsschema (Exchange, LK, Helsgaun, Simulated Annealing, evolutionäre Algorithmen,...) Kapitel 7. Ein Branch&Bound-Verfahren für das ATSP Assignment-B&B Kapitel 8. 1-Bäume, Lagrange-Relaxierung und untere Schranken durch ein Subgradientenverfahren (Held&Karp) Kapitel 9. Alternative IP-Modelle Kapitel 10. Das symmetrische TSP-Polytop Kapitel 11. Schnittebenenerkennung/Separationsalgorithmen

107

  

Vorlesungsplan

Kapitel 12. Zur praktischen Lösung großer TSPs Kapitel 13. TSPs mit Nebenbedingungen (Reihenfolgebedingungen, Zeitfenster, Multi-Salesmen,...) Unterwegs einbauen: Malen nach Zahlen, TSP-Portraits (Gesichter), Knights-Problem im Schach, Routenplanung,...

Martin Grötschel

The Travelling Salesman Problem a brief survey Martin Grötschel

Vorausschau auf die Vorlesung

The END

14. Oktober 2013

Martin Grötschel [email protected]

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