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Martin Grötschel Institute of Mathematics, Technische Universität Berlin (TUB) DFG-Research Center “Mathematics for key technologies” (MATHEON) Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB) [email protected] http://www.zib.de/groetschel
The Travelling Salesman Problem:
A brief survey
Martin GrötschelVorausschau auf die Vorlesung
Das Travelling-Salesman-Problem (ADM III)
im WS 2013/14
14. Oktober 2013
Martin Grötschel
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
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).
An important issue: How is I given?
Combinatorial optimization
I
min (I) ( ) | I , 2E
e
c c e I where I and E finite
Martin Grötschel
5Special „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.
Martin Grötschel
6Special „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 asymmetric (ATSP).
http://www.tsp.gatech.edu/
Martin Grötschel
9
THE TSPbook
suggested reading for everyone interested in the TSP
Another recommendationBill Cook‘s new book
Martin Grötschel
10
Martin Grötschel
11
The travelling salesman problem
1. :
( , )
. H
( ) .
min{ ( ) | H}.
2. :
{1,..., }
n
e
n
Version
Let K V E be the complete graph or digraphwith n nodes
and let c be the length of e E Let be the set of all
hamiltonian cycles tours in K Find
c T T
Version
Find a cyclic permutation of n such that
c
( )1
.
n
i ii
is as small as possible
Two mathematical formulations of the TSP
Does that help solve the TSP?
Martin Grötschel
12
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
13
Usually quoted as
the forerunner of
the TSP
Usually quoted as
the origin of
the TSP
about 100yearsearlier
Martin Grötschel
15
By a proper choice andscheduling of the tour onecan gain so much time that we have to makesome suggestions
The most important aspect is to cover as many locations as possiblewithout visiting alocation twice
A TSP contest 1962: 10.000 $ Prize
Martin Grötschel
16
Martin Grötschel
17
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
18
Ulysses
The distance table
Martin Grötschel
19
Ulysses roundtrip
optimal „Ulysses tour“
Martin Grötschel
20
Malen nach ZahlenTSP in art ?
When was this invented?
Martin Grötschel
21
Survey BooksLiterature: 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
22
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
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
24http://www.densis.fee.unicamp.br/~moscato/TSPBIB_home.html
Martin Grötschel
25
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
26
Production of ICs and PCBs
Integrated Circuit (IC) Printed Circuit Board (PCB)
Problems: Logical Design, Physical DesignCorrectness, Simulation, Placement of
Components, Routing, Drilling,...
Martin Grötschel
27Correct modelling of a printed circuit board drilling problem
2103 holes to be drilled
length of a move of the drilling head:Euclidean norm,Max norm,Manhatten norm?
Martin Grötschel
28
Drilling 2103 holes into a PCB
Significant Improvementsvia TSP
(due to Padberg & Rinaldi)
industry solution optimal solution
Siemens-ProblemPCB da4
before after
Martin Grötschel, Michael Jünger, Gerhard Reinelt,Optimal Control of Plotting and Drilling Machines: A Case Study, Zeitschrift für Operations Research, 35:1 (1991) 61-84http://www.zib.de/groetschel/pubnew/paper/groetscheljuengerreinelt1991.pdf
Siemens-Problem PCB da1
before after
Grötschel, Jünger, Reinelt
Martin Grötschel
31
Martin Grötschel
32Leiterplatten-BohrmaschinePrinted 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
da1 da2 da3 da4
Number of holes
Number of drills
Tour length
2457
7
3518728
423
7
1049956
2203
6
1958161
2104
10
4347902
Table 4
Martin Grötschel
37
Fast heuristics
da1 da2 da3 da4
CPU time (min:sec)
Tour length
Improvement in %
1:58
1695042
56.87
0:05
984636
14.60
1:43
1642027
26.94
1:43
1928371
58.38
Table 5
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
Andrea Grötschel Diplomarbeit (2004)
Logistics of collectingelectronics garbage
Martin Grötschel
42 Location plus tour planning (m-TSP)
Martin Grötschel
43
The Dispatching Problem at ADAC:an online m-TSP
Dispatching Center (Pannenzentrale)
Dispatcher
Data Transm.
„Gelber Engel“
Online-TSP (in a metric space)
Instance:
1 2, , , nr r r ( , )i i ir t xwhere
1x
1t t0
1x
2t t0
2x
Goal:Find fastest tour serving all requests (starting and ending in 0)
Algorithm ALG is c-competitive if
ALG OPTc for all request sequences
Martin Grötschel
45
Implementation competitions
Martin Grötschel
46
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
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
min
( ( )) 1 0
( ( )) 1 0
( ( )) | | 1 0 ,2 | |
0,1 ( , ) .
T
ij
c x
x i i V
x i i V
x A W W W V W n
x i j A
IP formulation of the asymmetric TSP
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 processesMartin Grötschel
49
Martin Grötschel
50
min
( ( )) 1 0
( ( )) 1 0
(1 ) , , 0
0
0,1 , .
T
i ij ij j
i i i
i
ij
c x
x i i V
x i i V
t x M t i j A j
r t d i V
t i V
x i j A
N
Model 1
Martin Grötschel
51
1 2 )
min
( ( )) 1 0
( ( )) 1 0
( ( )) | | 1 0 ,2 | |
( ) | | 1 2 infeasible path ( , , ,
0,1 ( , ) .
T
k
ij
c x
x i i V
x i i V
x A W W W V W n
x P P k P v v v
x i j A
Model 2
Martin Grötschel
52
1 0 1
min
( ( )) 1 0
( ( )) 1 0
, 0, , , , 0
0,1 ( , )
0,1,2,... ( , )
T
n n n
ij ij ij jki i ki j i j k j
i ij ij i ij
ij
ij
c x
x i i V
x i i V
y x y j V
r x y d x i j n i j i
x i j A
y i j A
Model 3
min
( ( )) 1 0
( ( )) 1 0
(1 ) , , 0
0
0,1 , .
T
i ij ij j
i i i
i
ij
c x
x i i V
x i i V
t x M t i j A j
r t d i V
t i V
x i j A
N
Model 1, 2, 3
1 2 )
min
( ( )) 1 0
( ( )) 1 0
( ( )) | | 1 0 ,2 | |
( ) | | 1 2 infeasible path ( , , ,
0,1 ( , ) .
T
k
ij
c x
x i i V
x i i V
x A W W W V W n
x P P k P v v v
x i j A
1 0 1
min
( ( )) 1 0
( ( )) 1 0
, 0, , , , 0
0,1 ( , )
0,1,2,... ( , )
T
n n n
ij ij ij jki i ki j i j k j
i ij ij i ij
ij
ij
c x
x i i V
x i i V
y x y j V
r x y d x i j n i j i
x i j A
y i j A
Martin Grötschel
54
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
55
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
56
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
57
Unevenness of Computational Results
problem #nodes gap #cutting planes
#LPs time
rbg041a
43 9.16% > 1 mio 109,402 > 5 h
rbg067a
69 0% 176 2 6 sec
Largest problem solved to optimality: 127 nodesLargest problem not solved optimally: 43 nodes
Martin Grötschel
58
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
59
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
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).
Minimization:
dual heuristic value ≤ optimum value ≤ primal heuristic value
quality guaranteein practice and theory
Martin Grötschel
61
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
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)
…..
The typical heuristics junk
Hyper-heuristics in Co-operative SearchThe 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
63
Martin Grötschel
64
Heuristics: A SurveyCurrently 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 analysisworst case analysis
There is no polynomial time approx. algorithm for STSP/ATSP. Christofides algorithm for the STSP with triangle inequality
average case analysisaverage case analysis Karp‘s analysis of the patching algorithm for the ATSP
probabilistic problem analysisprobabilistic problem analysis for Euclidean STSP in unit square, TSP constant 1.714..
polynomial time approximation schemes (PAS)polynomial time approximation schemes (PAS) Arora‘s polynomial-time approximation schemes for
Euclidean STSPs fully-polynomial time approximation schemes (FPAS)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 evaluationexperimental evaluation Lin-Kernighan for STSP (DIMACS challenges))
n
Martin Grötschel
66
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
67
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
68
The symmetric travelling salesman polytope
|: { } ( 1 , 0)
{ | ( ( )) 2
( ( )) | | 1 \ 1 ,3 | | 3
0 1 }
min
( ( )) 2
( ( )) | | 1 \ 1 ,3 | | 3
0,1
n T E TT n ij
E
ij
T
ij
T tour in KQ conv if ij T else
x x i i V
x E W W W V W n
x ij E
c x
x i i V
x E W W W V W n
x ij E
Z
R
The LP relaxation is solvable in polynomial time
Martin Grötschel
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
General cutting plane theory:Gomory Mixed-Integer Cut Given and
Rounding: Where define
Then
Disjunction:
Combining
, ,jy x ¢
, 0ij jy a x d d f f ,ij ij ja a f
: :ij j j ij j jt y a x f f a x f f ¢
: 1 :j j j j j jf x f f f x f f d t
:
1 : 1
j j j
j j j
t d f x f f f
t d f x f f f
: 1 1 : 1j j j j j jf f x f f f f x f f
clique trees
A clique tree is a connected graph C=(V,E), composed of cliques satisfying the following properties
Martin Grötschel
71
Martin Grötschel
72
Polyhedral Theory of the TSPComb inequality
2-matchingconstraint
toothhandle
Martin Grötschel
73
Clique Tree Inequalities
Martin Grötschel
74 Clique Tree Inequalities
http://www.zib.de/groetschel/pubnew/paper/groetschelpulleyblank1986.pdf
1 1 1
1 1 1 1
( ( )) ( ( )) | | 2
1( ( )) ( ( )) | | (| | )
2
h h
i i
h h
i
i j i
i
t
j i i
j
t
j ji
t
jT
H
H
H hT t
tT t
x
x H
x
E x E
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
75
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 etc.
Martin Grötschel
76
A very special case
Petersen graph, G = (V, F),
the smallest hypohamiltonian graph
10( ) 9
, 11
T
nT
x F defines a facet of Q
but not a facet of Q n
M. Grötschel & Y. Wakabayashi
Hypotraceable graphs and the STSP
On the right is the smallestknown hypotraceable graph(Thomassen graph, 34 nodes).
Such graphs have no hamiltonian path, but when any node is deleted, theremaining graph has ahamiltonian path.
How do such graphs induceinequalities valid for thesymmetric travelling salesmanpolytope?
For further information see:http://www.zib.de/groetschel/pubnew/paper/groetschel1980b.pdfMartin
Grötschel
77
“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.
Information “hypohamiltonian” and “hypotraceable” inequalities can be found inhttp://www.zib.de/groetschel/pubnew/paper/groetschelwakabayashi1981a.pdfhttp://www.zib.de/groetschel/pubnew/paper/groetschelwakabayashi1981b.pdf
Martin Grötschel
78
Martin Grötschel
79Valid 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
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
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 optimizationare 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
The LOWER BOUND is the mathematically andalgorithmically hard part of the work
Martin Grötschel
87
Work on LP relaxations of the symmetric travelling salesman polytope
|: { }
min
( ( )) 2
( ( )) | | 1 \ 1 ,3 | | 3
0 1
0,1
n T ET n
T
ij
ij
T tour in KQ conv
c x
x i i V
x E W W W V W n
x ij E
x ij E
Z
Integer Programming Approach
Martin Grötschel
88
cutting plane technique for integer and mixed-integer programming
Feasibleintegersolutions
LP-based relaxation
Convex hull
Objectivefunction
Cuttingplanes
Martin Grötschel
89
Clique-tree cut for pcb442 from B. Cook
Martin Grötschel
90
LP-based Branch & Bound
Root
Integer
v =0 v =1
x =
0 x =1
y =0
y =1
z =0 z = 1
Lower Bound
Integer
Upper Bound
Infeas
z =
0z = 1
GAP
GAP
Remark: GAP = 0 Proof of optimality
Solve LP relaxation: v=0.5 (fractional)
Martin Grötschel
91
A BranchingTree
Applegate
Bixby
Chvátal
Cook
Martin Grötschel
92
Managing the LPs of the TSP
CORE LP
astr
onom
ical
|V|(|V|-1)/2C
uts:
Sep
arat
ion
~ 3|V| variables~1.5|V| constraints
Column generation: Pricing.
Martin Grötschel
93
A Pictorial History of Some TSP World Records
Martin Grötschel
94
Some TSP World Records
year authors # cities # variables
1954 DFJ 42/49 820/1,146
1977 G 120 7,140
1987 PR 532 141,246
1988 GH 666 221,445
1991 PR 2,392 2,859,636
1992 ABCC 3,038 4,613,203
1994 ABCC 7,397 27,354,106
1998 ABCC 13,509 91,239,786
2001 ABCC 15,112 114,178,716
2004 ABCC 24,978 311,937,753
number of cities2000x
increase
4,000,000times
problem sizeincrease
in 52years
2005 W. Cook, D. Epsinoza, M. Goycoolea 33,810 571,541,145
2006pla 85,900
solved3,646,412,050
variables
Martin Grötschel
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
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USA 49
49 cities1,146 variables
1954
G. Dantzig, D.R. Fulkerson, S. Johnson
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97West-Deutschland und
Berlin
120 Städte7140 Variable
1975/1977/1980
M. Grötschel
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A tour around the world666 cities
221,445 variables
1987/1991
M. Grötschel, O. Holland, seehttp://www.zib.de/groetschel/pubnew/paper/groetschelholland1991.pdf
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USA cities with population >500
13,509 cities
91,239,786Variables
1998
D. Applegate, R.Bixby, V. Chvátal, W. Cook
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usa13509: The branching tree
0.01% initial gap
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Summary: usa13509
9539 nodes branching tree
48 workstations (Digital Alphas, Intel Pentium IIs, Pentium Pros, Sun UntraSparcs)
Total CPU time: 4 cpu years
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Overlay of3 OptimalGermanytours
fromABCC 2001
http://www.math.princeton.edu/tsp/d15sol/dhistory.html
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Optimal Tour of Sweden
311,937,753
variables
ABCCplusKeld
HelsgaunRoskilde
Univ. Denmark.
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World Tour, current statushttp://www.tsp.gatech.edu/world/
We give links to several images of the World TSP tourof 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.
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-PolytopeSubtour-Formulierungen, STSP und ATSP-Polytop
Kapitel 4. Kombinatorische Verwandte des TSPDas 1-Baum-, 2-Matching-, Zuordnungs-, 1-Arboreszenz-Problem
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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 ATSPAssignment-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
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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,...
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Martin Grötschel Institute of Mathematics, Technische Universität Berlin (TUB) DFG-Research Center “Mathematics for key technologies” (MATHEON) Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB) [email protected] http://www.zib.de/groetschel
The Travelling Salesman Problem
a brief survey
Martin GrötschelVorausschau auf die Vorlesung
Das Travelling-Salesman-Problem (ADM III)
14. Oktober 2013
The END