PS 3401 the Traveling Salesman Problem

8/4/2019 PS 3401 the Traveling Salesman Problem

1/7

PS 3401 The Traveling Salesman Problem

Number of Tours

It is easy to calculate the number of different tours through n cities: given a starting city, we have n-1choices for the second city, n-2 choices for the third city, etc. Multiplying these together we get (n-1)!

= n-1 x n-2 x n-3 x. . . x 3 x 2 x 1. Now since our travel costs do not depend on the direction we take

around the tour, we should divide this number by 2 to get (n-1)!/2.

This is a very large number (the actual value forn = 3038 is given below) and it is often cited as thereason the TSP seems to be so difficult to solve. It is true that the rapidly growing value of(n-1)!/2

rules out the possibility of checking all tours one by one, but there are other problems that are easy to

solve (such as the minimum spanning tree) where the number of solutions forn points grows evenmore quickly.

Number of Tours for a 3,038-City TSP

1179738538 7515997567 2260119227 7856391456 0323307555 1203849566

6223857826 1179738538 7515997567 2260119227 7856391456 0323307555

1203849566 6223857826 3070636136 5215602746 9206644453 5541847325

1


2/7

2220130370 7539740447 2752122520 1291614091 2145091403 9732067302

9095006463 8027485889 3957751434 6362958528 3224729785 7733992121

The History of Traveling salesman problem

Mathematical problems related to the traveling salesman problem were treated in the 1800s by the Irish

mathematician Sir William Rowan Hamilton and by the British mathematician Thomas PenyngtonKirkman. The picture below is a photograph of Hamilton's Icosian Game that requires players to

complete tours through the 20 points using only the specified connections. A nice discussion of the

early work of Hamilton and Kirkman can be found in the bookGraph Theory 1736-1936by N. L.Biggs, E. K. LLoyd, and R. J. Wilson, Clarendon Press, Oxford, 1976.

The general form of the TSP appears to be have been first studied by mathematicians starting in the

1930s by Karl Mengerin Vienna and Harvard. The problem was later promoted by Hassler Whitney

and Merrill Flood at Princeton. A detailed treatment of the connection between Menger and Whitney,

and the growth of the TSP as a topic of study can be found inAlexander Schrijver's paper ``On thehistory of combinatorial optimization (till 1960)''.

Pictorial History of the TSP

Milestones in the Solution of TSP Instances TSP Annotated Bibliography

Traveling versus Travelling

2
http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Hamilton.htmlhttp://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Kirkman.htmlhttp://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Kirkman.htmlhttp://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Kirkman.htmlhttp://www.oup.co.uk/isbn/0-19-853916-9http://www.oup.co.uk/isbn/0-19-853916-9http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Menger.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Whitney.htmlhttp://infoshare1.princeton.edu:2003/libraries/firestone/rbsc/finding_aids/mathoral/pmc11.htmhttp://www.cwi.nl/~lex/http://www.cwi.nl/~lex/http://www.cwi.nl/~lex/http://www.cwi.nl/~lex/files/histco.pshttp://www.cwi.nl/~lex/files/histco.pshttp://www.cwi.nl/~lex/files/histco.pshttp://www.tsp.gatech.edu/history/pictorial/pictorial.htmlhttp://www.tsp.gatech.edu/history/milestone.htmlhttp://www.tsp.gatech.edu/history/biblio/tspbiblio.htmlhttp://www.tsp.gatech.edu/history/travelling.htmlhttp://www-groups.dcs.st-andrews.ac.uk/~history/Diagrams/Icosian_game.jpeghttp://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Hamilton.htmlhttp://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Kirkman.htmlhttp://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Kirkman.htmlhttp://www.oup.co.uk/isbn/0-19-853916-9http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Menger.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Whitney.htmlhttp://infoshare1.princeton.edu:2003/libraries/firestone/rbsc/finding_aids/mathoral/pmc11.htmhttp://www.cwi.nl/~lex/http://www.cwi.nl/~lex/files/histco.pshttp://www.cwi.nl/~lex/files/histco.pshttp://www.tsp.gatech.edu/history/pictorial/pictorial.htmlhttp://www.tsp.gatech.edu/history/milestone.htmlhttp://www.tsp.gatech.edu/history/biblio/tspbiblio.htmlhttp://www.tsp.gatech.edu/history/travelling.html


3/7

3


4/7

First Application

Genome Sequencing

4


5/7

Researchers at the National Institute of Health have used Concorde's TSP solver to construct radiation

hybrid maps as part of their ongoing work in genome sequencing. The TSP provides a way tointegrate local maps into a single radiation hybrid map for a genome; the cities are the local maps and

the cost of travel is a measure of the likelihood that one local map immediately follows another. A

report on the work is given in the paper "A Fast and Scalable Radiation Hybrid Map Construction and

Integration Strategy", by R. Agarwala, D.L. Applegate, D. Maglott, G.D. Schuler, and A.A. Schaffler.

This application of the TSP has been adopted by a group in France developing a map of the mouse

genone. The mouse work is decribed in "A Radiation Hybrid Transcript May of the Mouse Genome",

Nature Genetics29 (2001), pages 194--200.

Next Application

Starlight Interferometer Program

A team of engineers at Hernandez Engineering in Houston and at Brigham Young University have

experimented with using Chained Lin-Kerninghan to optimize the sequence of celestial objects to beimaged in a proposed NASA Starlightspace interferometer program. The goal of the study it to

minimize the use of fuel in targeting and imaging maneuvers for the pair of satellites involved in the

mission (the cities in the TSP are the celestial objects to be imaged, and the cost of travel is the amountof fuel needed to reposition the two satellites from one image to the next). A report of the work is

given in the paper "Fuel Saving Strategies for Separated Spacecraft Interferometry".

Next Application

Scan Chain Optimization

A semi-conductor manufacturer has used Concorde's implementation of the Chained Lin-Kernighanheuristic in experiments to optimize scan chains in integrated circuits. Scan chains are routes included

on a chip for testing purposes and it is useful to minimize their length for both timing and power

reasons.

5
http://www.ncbi.nlm.nih.gov/genome/rhmap/http://www.ncbi.nlm.nih.gov/genome/rhmap/http://www.nature.com/ng/http://www.nature.com/ng/http://www.tsp.gatech.edu/apps/starlight.htmlhttp://www.ee.byu.edu/grad1/users/beard/www_docs/research_formation.htmlhttp://www.tsp.gatech.edu/apps/scan.htmlhttp://www.ncbi.nlm.nih.gov/genome/rhmap/http://www.ncbi.nlm.nih.gov/genome/rhmap/http://www.nature.com/ng/http://www.tsp.gatech.edu/apps/starlight.htmlhttp://www.ee.byu.edu/grad1/users/beard/www_docs/research_formation.htmlhttp://www.tsp.gatech.edu/apps/scan.html


6/7

Next Application

DNA Universal Strings

A group at AT&T used Concorde to compute DNA sequences in a genetic engineering researchproject. In the application, a collection of DNA strings, each of length k, were embedded in one

universal string (that is, each of the target strings is contained as a substring in the universal string),

with the goal of minimizing the length of the universal string. The cities of the TSP are the target

strings, and the cost of travel is kminus the maximum overlap of the corresponding strings.

Next Application

Whizzkids '96 Vehicle Routing

A modified version of Concorde was used to solve the Whizzkids'96 vehicle routing problem,demonstrating that the winning solution in the 1996 competition was in fact optimal. The problem

consists of finding the best collection of routes for 4 newsboys to deliver papers to their 120

customers. The team of David Applegate, William Cook, Sanjeeb Dash, and Andre Rohe received a5,000 Gulden prize for their solution in February 2001 from the information technology firm CMG.

Next Application

A Tour Through MLB Ballparks

A baseball fan found the optimal route to visit all 30 Major League Baseball parks using Concorde's

solver. The data for the problem can be found in the file mlb30.tsp (the data set is in TSPLIBformat).

The map above is a link to an interesting site devoted to current and past ballparks.

6
http://www.tsp.gatech.edu/apps/dna.htmlhttp://www.uark.edu/campus-resources/mivey/m4233/DNAstruc.htmhttp://www.tsp.gatech.edu/apps/whizzkids.htmlhttp://www.cmg.com/http://www.cmg.com/http://www.tsp.gatech.edu/apps/ballparks.htmlhttp://www.tsp.gatech.edu/apps/mlb30.tsphttp://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/http://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/http://www.ballparks.com/baseballhttp://www.win.tue.nl/whizzkids/1996http://www.gch.ulaval.ca/~agarnier/mol10.gifhttp://www.tsp.gatech.edu/apps/dna.htmlhttp://www.uark.edu/campus-resources/mivey/m4233/DNAstruc.htmhttp://www.tsp.gatech.edu/apps/whizzkids.htmlhttp://www.cmg.com/http://www.tsp.gatech.edu/apps/ballparks.htmlhttp://www.tsp.gatech.edu/apps/mlb30.tsphttp://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/


7/7

Next Application

Genome Sequencing

An old application of the TSP is to schedule the collection of coins from payphones throughout a given

region. A modified version of Concorde's Chained Lin-Kernighan heuristic was used to solve a variety

of coin collection problems. The modifications were needed to handle 1-way streets and other featuresof city-travel that make the assumption that the cost of travel from x to y is the same as from y to x

unrealistic in this scenario.

Next Application

Touring Airports

Concorde is currently being incorporated into the Worldwide Airport Path Finderweb site to find

shortest routes through selections of airports in the world. The author of the site writes that users of the

path-finding tools are equally split between real pilots and those using flight simulators.

Next Application

USA Trip

The travel itinerary for an executive of a non-profit organization was computed using Concorde's TSP

solver. The trip involved a chartered aircraft to visit cities in the 48 continental states plus

Washington, D.C. (Commercial flights were used to visit Alaska and Hawaii.) It would have beennice if the problem was the same as that solved in 1954 by Dantzig, Fulkerson, and Johnson, but

different cities were involved in this application (and somewhat different travel costs, since flight

distances do not agree with driving distances). The data for the instance was collected by Peter Winker

of Lucent Bell Laboratories.

7
http://www.tsp.gatech.edu/apps/coins.htmlhttp://www.tsp.gatech.edu/apps/airports.htmlhttp://www.fallingrain.com/air/http://www.tsp.gatech.edu/apps/usatrip.htmlhttp://www.telephoneart.com/pay/dbped2a.jpghttp://www.tsp.gatech.edu/apps/coins.htmlhttp://www.tsp.gatech.edu/apps/airports.htmlhttp://www.fallingrain.com/air/http://www.tsp.gatech.edu/apps/usatrip.html

PS 3401 the Traveling Salesman Problem

Documents

Transcript of PS 3401 the Traveling Salesman Problem