The Travelling Salesman Problem: A brief survey

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

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Contents1. Introduction2. The TSP and some of its history3. The TSP and some of its variants4. Some applications5. Modeling issues6. Heuristics7. How combinatorial optimizers do it

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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 , 2Ee

c c e I where I and E finite

Martin Grötschel

5Special „simple“ combinatorial optimization problemsFinding 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.

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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).

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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/

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THE TSPbook

suggested reading for everyone interested in the TSP

Another recommendationBill Cook‘s new book

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11 The travelling salesman problem

1. :( , )

. H( ) .

min{ ( ) | H}.

2. :{1,..., }

n

e

n

VersionLet K V E be the complete graph or digraphwith n nodesand let c be the length of e E Let be the set of allhamiltonian cycles tours in K Find

c T T

VersionFind 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?

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Usually quoted as the forerunner of the TSP

Usually quoted as the origin of the TSP

about 100yearsearlier

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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

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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

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Ulysses

The distance table

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Ulysses roundtrip

optimal „Ulysses tour“

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Malen nach ZahlenTSP in art ?

When was this invented?

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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

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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

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

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Production of ICs and PCBs

Integrated Circuit (IC) Printed Circuit Board (PCB)

Problems: Logical Design, Physical DesignCorrectness, Simulation, Placement of

Components, Routing, Drilling,...

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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?

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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

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32Leiterplatten-BohrmaschinePrinted Circuit Board Drilling Machine

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Foto einer Flachbaugruppe (Leiterplatte)

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Foto einer Flachbaugruppe (Leiterplatte) - Rückseite

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442 holes to be drilled

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Typical PCB drilling problems at Siemens

da1 da2 da3 da4

Number of holesNumber of drillsTour length

24577

3518728

4237

1049956

22036

1958161

210410

4347902

Table 4

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Fast heuristics

da1 da2 da3 da4

CPU time (min:sec)Tour length

Improvement in %

1:581695042

56.87

0:05984636

14.60

1:431642027

26.94

1:431928371

58.38

Table 5

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Optimizing the stacker cranes of a Siemens-Nixdorf warehouse

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Herlitz at Falkensee (Berlin)

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Example: Control of the stacker cranes in a Herlitz

warehouse

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Andrea Grötschel Diplomarbeit (2004)

Logistics of collectingelectronics garbage

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42 Location plus tour planning (m-TSP)

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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

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Implementation competitions

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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

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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 processes

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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

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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 nx P P k P v v v

x i j A

Model 2

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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

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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

, ,( , )

,

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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

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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?

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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

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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 ?

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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

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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

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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.

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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

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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

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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

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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 Vx E W W W V W n

x ij E

c xx i i Vx E W W W V W n

x ij E

ZR

The LP relaxation is solvable in polynomial time

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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)

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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

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Polyhedral Theory of the TSPComb inequality

2-matchingconstraint

toothhandle

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Clique Tree Inequalities

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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 handlesTj, j=1,…,t are the teethtj is the number of handles that tooth Tj intersects

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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.

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A very special case

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

10( ) 9

, 11T

nT

x F defines a facet of Q

but not a facet of Q n

M. Grötschel & Y. Wakabayashi

Hypotraceable graphs and the STSPOn 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

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“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

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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)

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Counting Tours and Facetsn # tours # different facets # facet classes3 1 0 04 3 3 15 12 20 26 60 100 47 360 3,437 68 2520 194,187 249 20,160 42,104,442 192

10 181,440 >= 52,043,900,866 >=15,379

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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?

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Separation

K

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

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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

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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.

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Deutschland 15,112

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

15,112 cities

114,178,716variables

2001

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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

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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 xx i i Vx E W W W V W n

x ij E

x ij E

Z

Integer Programming Approach

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cutting plane technique for integer and mixed-integer programming

Feasibleintegersolutions

LP-based relaxation

Convex hull

Objectivefunction

Cuttingplanes

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Clique-tree cut for pcb442 from B. Cook

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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 = 0

z = 1

GAP

Remark: GAP = 0 Proof of optimality

Solve LP relaxation: v=0.5 (fractional)

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A BranchingTree

ApplegateBixbyChvátalCook

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Managing the LPs of the TSP

CORE LP

astro

nom

ical

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

uts:

Sep

arat

ion ~ 3|V| variables

~1.5|V| constraints

Column generation: Pricing.

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93 A Pictorial History of Some TSP World Records

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94 Some TSP World Recordsyear authors # cities # variables1954 DFJ 42/49 820/1,1461977 G 120 7,1401987 PR 532 141,2461988 GH 666 221,4451991 PR 2,392 2,859,6361992 ABCC 3,038 4,613,2031994 ABCC 7,397 27,354,1061998 ABCC 13,509 91,239,7862001 ABCC 15,112 114,178,7162004 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

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The current championsABCC stands forD. 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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99 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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104 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/SeparationsalgorithmenMartin Grötschel

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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,...

Martin Grötschel

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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

The Travelling Salesman Problem: A brief survey

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