Applications of Discrete Optimization - The Swiss AI Lab IDSIA

330 Pages·2006·5.88 MB·English

Università della Svizzera italiana IDSIA
Istituto Dalle Molle di studi
Scuola universitaria professionale sull’intelligenza artificiale
della Svizzera italiana
Applications of Discrete
Optimization
Course at Department of Informatics and Electronics,
Politecnico di Milano, Second Semester, 2005-2006
Prof. Luca Maria Gambardella
IDSIA, Istituto Dalle Molle di Studi sull’Intelligenza Artificiale
Manno, Lugano, Switzerland
www.idsia.ch
Applications of Discrete Optimization, Gambardella, 2006, 1
Università della Svizzera italiana IDSIA
Istituto Dalle Molle di studi
Scuola universitaria professionale sull’intelligenza artificiale
della Svizzera italiana
Research Institute (~30 people) in Lugano since 1988
Basic Research
(Swiss National Science Foundation)
Optimization, Machine Learning,
(cid:132)
Bio-Inspired Algorithms, Artificial Neural Networks
(cid:132)
Business week in 1997 classified IDSIA among the
(cid:132)
best 10 worldwide AI institutes
Applied Research
(CTI, European Commission, Companies)
Optimization in transport (multimodal terminals, fleet
(cid:132)
of vehicles) and production.
Data Mining
(cid:132)
Applications of Discrete Optimization, Gambardella, 2006, 2
Contents
Travelling Salesman problems
models
exact algorithms
constructive techniques
meta-heuristics
Sequential ordering problem
meta-heuristics
Vehicle routing problems
models
exact algorithms
constructive techniques
meta-heuristics
industrial applications
Applications of Discrete Optimization, Gambardella, 2006, 3
TSP: Traveling Salesman Problems
Problem: given N cities, and a distance function d between
cities (usually time or kilometres), find a tour that:
goes through every city
(cid:132)
once and only once
Seattle
minimizes the total distance
(cid:132)
Boston
San Salt Lake City
Francisco
Las Vegas Indianapolis New York
Oklahoma
Los Angeles City
San Diego Phoenix Albuquerque Atlanta
Houston
Miami
Applications of Discrete Optimization, Gambardella, 2006, 4
Traveling Salesman Problems
A game as first TSP example
Hamilton’s Icosian Game
(1800)
It is required to
complete a tour along
20 points with a
restricted number of
connections
Hamilton’s Iconsian game
Applications of Discrete Optimization, Gambardella, 2006, 5
TSP. An easy problem?
Direct solution
Try all the possible permutations (ordered
combinations) and see which one is the cheapest
(using brute force)
The number of permutations is n! (factorial on
the number of cities, n)
The problem is NP-Hard
Applications of Discrete Optimization, Gambardella, 2006, 6
TSP history
• First description in 1800 by the Irish mathematician Sir
William Rowan Hamilton and the British mathematician
Thomas Penyngton Kirkman.
• The general form is presented for the first time in the
mathematic studies in 1930 by Karl Menger in Vienna and
Harvard. The problem was also promoted by Whitney and
Merrill Flood a Princeton.
• A detailed description of Menger and Whitney work and
of TSP diffusion can be found in Alexander Schrijver “On
the history of combinatorial optimization”, 1960.
Applications of Discrete Optimization, Gambardella, 2006, 7
TSP History
(cid:127) A breakthrough by George Dantzig, Ray
Fulkerson, and Selmer Johnson in 1954.
(cid:127) 49 - 120 – 550 - 2,392 - 7,397 – 19,509 cities.
From year 1954 to year 2001.
(cid:127) 24,098 cities by David Applegate, Robert Bixby,
Vasek Chvatal, William Cook, and Keld Helsgaun
in May 2004.
Applications of Discrete Optimization, Gambardella, 2006, 8
TSP instances
years Research team Problem size
1954 G.Dantzig, R. Fulkerson, and S. Johnson 49 cities
1971 M. Held and R.M.Karp 64 cities
1975 P.M.Camerini, L. Fratta, and F. Maffioli 100 cities
1977 M.Grötschel 120 cities
1980 H.Crowder and M.W.Padberg 318 cities
1987 M.Padbergand G.Rinaldi 532 cities
1987 M.Grötschel and O.Holland 666 cities
1987 M. Padbergand G.Rinaldi 2.392 cities
1994 D.Applegate, R.Bixby, V.Chvàtal, e W.Cook 7.397 cities
1998 D.Applegate, R.Bixby, V.Chvàtal, e W.Cook 13.509 cities
2001 D.Applegate, R.Bixby, V.Chvàtal, e W.Cook 15.112 cities
2004 D.Applegate, R.Bixby, V.Chvàtal, e W.Cook 24.978 cities
Applications of Discrete Optimization, Gambardella, 2006, 9
1954
G.Dantzig, R. Fulkerson, and S. Johnson
49 città
Applications of Discrete Optimization, Gambardella, 2006, 10