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 GrötschelSummary of Chapter 2

of the classPolyhedral Combinatorics (ADM III)

May 18, 2010

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




5. Modeling issues

6. Heuristics


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

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

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 aysmmetric (ATSP).

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

Martin Grötschel

9

THE TSPbook

suggested reading for everyone interested in the TSP

Martin Grötschel

10

The travelling salesman problem

1. :

( , )

.

( ) .

min{ ( ) | }.

2. :

{1,..., }

n

e

n

i

Version

Let K V E be the complete graph digraphwith n nodes

and let c be the length of e E Let H be the set of all

hamiltonian cycles tours in K Find

c T T H

Version

Find a cyclic permutation of n such that

c

( )1

.

n

ii

is as small as possible

Two mathematical formulations of the TSP

Does that help solve the TSP?

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11

Contents

1. Introduction




5. Modeling issues

6. Heuristics


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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

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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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Contents

1. Introduction




5. Modeling issues

6. Heuristics


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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

Martin Grötschel

22http://www.densis.fee.unicamp.br/~moscato/TSPBIB_home.html

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Contents

1. Introduction




5. Modeling issues

6. Heuristics


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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,...

Martin Grötschel

25Correct 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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26

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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29

Martin Grötschel

30Leiterplatten-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 holes

Number of drills

Tour length

2457

7

3518728

423

7

1049956

2203

6

1958161

2104

10

4347902

Table 4

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35

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

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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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40 Location plus tour planning (m-TSP)

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41

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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Contents

1. Introduction




5. Modeling issues

6. Heuristics


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

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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 processesMartin Grötschel

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Martin Grötschel

48

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

49

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

50

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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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

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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?

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55

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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Contents

1. Introduction




5. Modeling issues

6. Heuristics


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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 ?

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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 guaranteein practice and theory

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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

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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)

…..

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61

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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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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Contents

1. Introduction




5. Modeling issues

6. Heuristics


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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

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65

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

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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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67

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

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Martin Grötschel

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Polyhedral Theory of the TSPComb inequality

2-matchingconstraint

toothhandle

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Clique Tree Inequalities

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71 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

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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 etc.

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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

74

“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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75

Martin Grötschel

76Valid 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 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

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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

114,178,716

variables

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 x

x i i V

x 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 =

0z = 1

GAP

GAP

Remark: GAP = 0 Proof of optimality

Solve LP relaxation: v=0.5 (fractional)

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A BranchingTree

Applegate

Bixby

Chvátal

Cook

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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.

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A Pictorial History of Some TSP World Records

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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

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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

Martin Grötschel

94West-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.

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 GrötschelSummary of Chapter 2

of the classPolyhedral Combinatorics (ADM III)

May 18, 2010

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The Travelling Salesman Problem a brief survey

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