50+ Years of
Combinatorial
Integer
Programming
William Cook
50+ Years of
Combinatorial
Integer
Programming
Robert Bosch, December, 2007
Newsweek, July 26, 1954
Newsweek, July 26, 1954
Newsweek, July 26, 1954
“By an ingenious application of linear
programming -- a mathematical tool
recently used to solve productionscheduling problems -- it took only a
few weeks for the California experts to
calculate ‘by hand’ the shortest route ...”
Dantzig,
Fulkerson,
Johnson
OR 2 (1954)
Dantzig,
Fulkerson,
Johnson
OR 2 (1954)
“The origin of this problem is somewhat obscure.
It appears to have been discussed informally among
mathematicians at mathematics meetings for many
OR 2 (1954)
years.”
Karl Menger
Vienna
February 5, 1930
“We use the term Botenproblem (because this question
is faced in practice by every postman, and, by the way,
also by many travelers) for the task, given a finite
number of points with known pairwise distances, to find
the shortest path connecting the points.”
OR 2 (1954)
“Both Flood and A. W. Tucker (Princeton
University) recall that they heard about the
OR 2 (1954)
problem first in a seminar talk by Hassler
Whitney at Princeton in 1934 (although
Whitney, recently queried, does not seem to
recall the problem).”
Letter from A. Tucker
to D. Shmoys, 1983
An Interview of Merrill Flood with Albert Tucker
May 14, 1984
M. Flood (1984): “. . . the `48 States Problem’ of Hassler Whitney . . . I don’t know
who coined the peppier name `Traveling Salesman Problem’ for Whitney’s
problem . . .”
A. J. Hoffman and P. Wolfe (1985): “This would seem to make Whitney the
entrepreneur of the TSP---possibly as a messenger from Menger---but Whitney
remembers no connection whatever between himself and the TSP. Our inclination,
nevertheless, is to rely on Flood’s memory, even if Tucker and Whitney cannot
confirm it.”
A. Schrijver (2003): “There are some uncertainties in this story. It is not sure if
Whitney spoke about the 48 States problem during his 1931-1932 seminar talks
(which talks he did give), or later, in 1934, as is said by Flood [1956] . . .”
Title: Papers of Hassler Whitney, ca. 1930-1987
Description: 3.9 cubic feet in 12 containers
Handwritten notes,
1931 (?)
Pusey Library, Harvard Yard
“Given a set of countries, is it possible to
travel through them in such a way that at
the end of the trip we have visited each
country exactly once?”
OR 2 (1954)
“The relations between the traveling-salesman problem
and
the
transportation
problem
of
linear
programming
OR 2 (1954)
appear to have been first explored by M. Flood, J.
Robinson, T. C. Koopmans, M. Beckmann, and later by I.
Heller and H. Kuhn.”
THE SUMMER MEETING IN KINGSTON
The fifty-eighth Summer Meeting and the thirty-fourth Colloquium of the American Mathematical Society were held a t Queen's
University and the Royal Military College, Kingston, Ontario,
Canada, from August 31 to September S, 19S3 in conjunction with
meetings of the Canadian Mathematical Congress, the Mathematical
Association of America, the Institute of Mathematical Statistics, and
the Economic Society.
Over 700 people registered for the meeting, including the following 377 members of the Society:
AMS 1953
C. R. Adams, R. B. Adams, J. G. Adshead, M. I. Aissen, B. E. Allen, E. B. Allen,
C. B. Allendoerfer, J. P. van Alstyne, S. A. Amitsur, A. G. Anderson, R. D. Anderson, R. G. Archibald, Richard Arens, E. L. Arnoff, H. E. Arnold, K. J. Arnold, Nachman Aronszajn, M. G. Arsove, M. C. Ayer, I. A. Barnett, C. F. Barr, J. H. Barrett,
L. K. Barrett, R. C. F. Bartels, R. G. Bartle, P. T. Bateman, W. R. Baum, Samuel
Beatty, E. G. Begle, Theodore Bennett, R. H. Bing, D. W. Blackett, David Blackwell, J. R. Blum, M. I. Blyth, Salomon Bochner, H. F. Bohnenblust W. M.
Boothby, R. C. Bose, Evelyn Boyd, C. B. Boyer, A. T. Brauer, G. U. Brauer, F. R.
Britton, F. E. Browder, F. H. Browneil, G. S. Bruton, F. J. H. Burkett, G. H. Butcher,
James William Butler, P. L. Butzer, W. R. Callahan, E. A. Cameron, J, W. Campbell,
L. L. Campbell, K. H. Carlson, A. B. Carson, Henri Cartan, C. R. Cassity, Lamberto
Cesari, Abraham Charnes, Randolph Church, R. V. Churchill, Paul Civin, C. J. Clark,
H. M. Clark, A. B. Clarke, H. E. Clarkson, D. E. Coffey, L. W. Cohen, Harvey Cohn,
R. H. Cole, A. J. Coleman, J. B. Coleman, Esther Comegys, R. M. Conkling, T. F.
Cope, A. H. Copeland, Sr., H. S. M. Coxeter, J. B. Crabtree, C. C. Craig, H. F. Cullen, A. B. Cunningham, H. B. Curry, A. E. Danese, J. M. Danskin, G. B. Dantzig,
R. A. Dean, R. Y. Dean, J. C. E. Dekker, D. B. DeLury, Douglas Derry, W. F.
Donoghue, Jr., C. H. Dowker, A. C Downing, D. W. Dunn, W. L. Duren, Jr., W. F.
Eberlein, H. W. Ellis, M. P. Emerson, H. P. Evans, R. L. Evans, Walter Feit,
H. H. Ferns, J. V. Finch, D. T. Finkbeiner, Irwin Fischer, E. E. Floyd, G. E. Forsythe, J. S. Frame, Evelyn Frank, H. D. Friedman, A. H. Frink, Orrin Frink, R. M.
Frisch, R. E. Fullerton, I. N. Gâl, I. S. Gâl, H. L. Garabedian, A. O. Garder, Boris
Garfinkel, H. M. Gehman, Irving Gerst, Murray Gerstenhaber, K. S. Ghent, Leonard
Gillman, Seymour Ginsburg, Sidney Glusman, Herbert Goertzel, Casper Goffman,
Michael Goldberg, Harry Gonshor, S. H. Gould, R. L. Graves, J. W. Green, Harold
Greenspan, J. A. Greenwood, F. L. Griffin, V. G. Grove, Simon Gruenzweig, Irwin
Gutman, P. R. Halmos, M. E. Hamstrom, J. F. Hannan, B. I. Hart, G. E. Hay,
Henry Helson, Meivin Henriksen, J. G. Herriot, I. N. Herstein, M. J. Herzberger,
Edwin Hewitt, T. H. Hildebrandt, J. G. Hocking, R. V. Hogg, F. E. Hohn, D. L. Holl,
M. P. Hollcroft, T. R. Hollcroft, W. A. Hurwitz, W. R. Hutcherson, Jack Indritz,
S. L. Isaacson, J. R. Isbell, Arno Jaeger, T. J. Jaramillo, R. L. Jeffery, W. E. Jenner,
Meyer Jerison, A. E. Johns, L. W. Johnson, R. E. Johnson, F. E. Johnston, B. W.
Jones, P. S. Jones, G. K. Kalisch, L. H. Kanter, Leo Katz, D. E. Kearney, M. E.
Kellar, J. B. Kelly, J. R. F. Kent, D. E. Kibbey, T. C. Koopmans, C. F. Kossack,
H. L. Krall, Max Kramer, Saul Kravetz, Solomon Kullback, O. E. Lancaster,
C. E. Langenhop, E. H. Larguier, J. W. Lawson, C. Y. Lee, H. L. Lee, A. B. Lehman,
SIS
. Browneil, G. S. Bruton, F. J. H. Burkett, G. H. Butcher,
Butzer, W. R. Callahan, E. A. Cameron, J, W. Campbell,
on, A. B. Carson, Henri Cartan, C. R. Cassity, Lamberto
andolph Church, R. V. Churchill, Paul Civin, C. J. Clark,
H. E. Clarkson, D. E. Coffey, L. W. Cohen, Harvey Cohn,
J. B. Coleman, Esther Comegys, R. M. Conkling, T. F.
H. S. M. Coxeter, J. B. Crabtree, C. C. Craig, H. F. CulB. Curry, A. E. Danese, J. M. Danskin, G. B. Dantzig,
. C. E. Dekker, D. B. DeLury, Douglas Derry, W. F.
er, A. C Downing, D. W. Dunn, W. L. Duren, Jr., W. F.
AMS 1953
P. Emerson,
H. P. Evans, R. L. Evans, Walter Feit,
D. T. Finkbeiner, Irwin Fischer, E. E. Floyd, G. E. ForFrank, H. D. Friedman, A. H. Frink, Orrin Frink, R. M.
N. Gâl, I. S. Gâl, H. L. Garabedian, A. O. Garder, Boris
ving Gerst, Murray Gerstenhaber, K. S. Ghent, Leonard
g, Sidney Glusman, Herbert Goertzel, Casper Goffman,
onshor, S. H. Gould, R. L. Graves, J. W. Green, Harold
d, F. L. Griffin, V. G. Grove, Simon Gruenzweig, Irwin
. E. Hamstrom, J. F. Hannan, B. I. Hart, G. E. Hay,
riksen, J. G. Herriot, I. N. Herstein, M. J. Herzberger,
THE SUMMER MEETING IN KINGSTON
The fifty-eighth Summer Meeting and the thirty-fourth Colloquium of the American Mathematical Society were held a t Queen's
University and the Royal Military College, Kingston, Ontario,
Canada, from August 31 to September S, 19S3 in conjunction with
meetings of the Canadian Mathematical Congress, the Mathematical
Association of America, the Institute of Mathematical Statistics, and
the Economic Society.
Over 700 people registered for the meeting, including the following 377 members of the Society:
C. R. Adams, R. B. Adams, J. G. Adshead, M. I. Aissen, B. E. Allen, E. B. Allen,
C. B. Allendoerfer, J. P. van Alstyne, S. A. Amitsur, A. G. Anderson, R. D. Anderson, R. G. Archibald, Richard Arens, E. L. Arnoff, H. E. Arnold, K. J. Arnold, Nachman Aronszajn, M. G. Arsove, M. C. Ayer, I. A. Barnett, C. F. Barr, J. H. Barrett,
L. K. Barrett, R. C. F. Bartels, R. G. Bartle, P. T. Bateman, W. R. Baum, Samuel
Beatty, E. G. Begle, Theodore Bennett, R. H. Bing, D. W. Blackett, David Blackwell, J. R. Blum, M. I. Blyth, Salomon Bochner, H. F. Bohnenblust W. M.
Boothby, R. C. Bose, Evelyn Boyd, C. B. Boyer, A. T. Brauer, G. U. Brauer, F. R.
Britton, F. E. Browder, F. H. Browneil, G. S. Bruton, F. J. H. Burkett, G. H. Butcher,
James William Butler, P. L. Butzer, W. R. Callahan, E. A. Cameron, J, W. Campbell,
L. L. Campbell, K. H. Carlson, A. B. Carson, Henri Cartan, C. R. Cassity, Lamberto
Cesari, Abraham Charnes, Randolph Church, R. V. Churchill, Paul Civin, C. J. Clark,
H. M. Clark, A. B. Clarke, H. E. Clarkson, D. E. Coffey, L. W. Cohen, Harvey Cohn,
R. H. Cole, A. J. Coleman, J. B. Coleman, Esther Comegys, R. M. Conkling, T. F.
Cope, A. H. Copeland, Sr., H. S. M. Coxeter, J. B. Crabtree, C. C. Craig, H. F. Cullen, A. B. Cunningham, H. B. Curry, A. E. Danese, J. M. Danskin, G. B. Dantzig,
R. A. Dean, R. Y. Dean, J. C. E. Dekker, D. B. DeLury, Douglas Derry, W. F.
Donoghue, Jr., C. H. Dowker, A. C Downing, D. W. Dunn, W. L. Duren, Jr., W. F.
Eberlein, H. W. Ellis, M. P. Emerson, H. P. Evans, R. L. Evans, Walter Feit,
H. H. Ferns, J. V. Finch, D. T. Finkbeiner, Irwin Fischer, E. E. Floyd, G. E. Forsythe, J. S. Frame, Evelyn Frank, H. D. Friedman, A. H. Frink, Orrin Frink, R. M.
Frisch, R. E. Fullerton, I. N. Gâl, I. S. Gâl, H. L. Garabedian, A. O. Garder, Boris
Garfinkel, H. M. Gehman, Irving Gerst, Murray Gerstenhaber, K. S. Ghent, Leonard
Gillman, Seymour Ginsburg, Sidney Glusman, Herbert Goertzel, Casper Goffman,
Michael Goldberg, Harry Gonshor, S. H. Gould, R. L. Graves, J. W. Green, Harold
Greenspan, J. A. Greenwood, F. L. Griffin, V. G. Grove, Simon Gruenzweig, Irwin
Gutman, P. R. Halmos, M. E. Hamstrom, J. F. Hannan, B. I. Hart, G. E. Hay,
Henry Helson, Meivin Henriksen, J. G. Herriot, I. N. Herstein, M. J. Herzberger,
Edwin Hewitt, T. H. Hildebrandt, J. G. Hocking, R. V. Hogg, F. E. Hohn, D. L. Holl,
M. P. Hollcroft, T. R. Hollcroft, W. A. Hurwitz, W. R. Hutcherson, Jack Indritz,
S. L. Isaacson, J. R. Isbell, Arno Jaeger, T. J. Jaramillo, R. L. Jeffery, W. E. Jenner,
Meyer Jerison, A. E. Johns, L. W. Johnson, R. E. Johnson, F. E. Johnston, B. W.
Jones, P. S. Jones, G. K. Kalisch, L. H. Kanter, Leo Katz, D. E. Kearney, M. E.
Kellar, J. B. Kelly, J. R. F. Kent, D. E. Kibbey, T. C. Koopmans, C. F. Kossack,
H. L. Krall, Max Kramer, Saul Kravetz, Solomon Kullback, O. E. Lancaster,
C. E. Langenhop, E. H. Larguier, J. W. Lawson, C. Y. Lee, H. L. Lee, A. B. Lehman,
SIS
be the intersection of Nh with C. A self-adjoint finite-difference operator Ah approximating A is introduced; it is the symmetric part of the operator (65.1) in W. E.
Milne, Numerical solution of differential equations, Wiley, 1953. Let X& be the least
number such that AfcV-f-XfcW — O in Rh, where v=*v(x, y) is defined for (x, y)ÇzRh*<JCh
and vanishes on Ch. Theorem: As *-*0, \h/\^l-h2(A+B)/D+o(h2).
Here A
2
^//RiuL+ulJdxdy;
B=Jcu\ sin 2rdr; D = \2ffR(ux+uy)dxdy', un is the normal
derivative of u; r is the angle between the tangent to u and the x axis; u is the fundamental solution of (*). Thus ultimately XAÎX. This extends the result for certain
polygonal regions R announced and discussed in Bull. Amer. Math. Soc. Abstract
59-4-509. (Received July 13, 1953.)
AMS 1953
664/. Isidor Heller: On the problem of shortest path between points. I.
The n\ closed paths connecting n given points are represented by the set P» of
n by n permutation matrices. These, interpreted as points in w2-space, are the extreme
points of the convex set of doubly stochastic matrices (Birkhoff, von Neumann).
In problem above only the subset Cn consisting of the (» —1)! cycles of order n is
admitted. Cn has dimension (n — l) 2 —n so that the 6 points of d form a 5-dimensional
simplex (H. Kuhn in a letter to author). A first objective was to determine the
extreme hyperplanes of the convex of Cg, which is an 11-dimensional polytope in
25-space with 24 vertices, the main difficulty opposing straightforward computation
being the nearly 2.5 million possibilities of choosing 11 points out of 24. Results: The
convex of Cg is characterized by a nonredundant system of 224 hyperplanes of the
following 6 types. (1) Xu*z0 (i^j); (2) X»i = 0; (3) sum of any row = sum of any
column = 1 (one of the 2n equations to be omitted in order to avoid redundancy) ;
(4) Xij+Xji^l;
(5) Xii+Xji+Xn-Xit-Xtrgl
for distinct (i, j , r, s, t)\ (6) IXa
WXji-Xir+Xir-Xsi+Xsj-Xrt^îovdistinct
(i,j,r,s). (ReceivedJune24, 1953.)
665/. Isidor Heller: On the problem of shortest path between points.
II.
J. Dessart [Sur les surfaces représentant Vinvolution engendré par un homography
de cinque du plan, Memoires de la Société Royale des Sciences de Liège no. 17 (1931)
pp. 1-23] and other writers have used the homography x[ :xi :xi —Xi:Ex2'*EaXz where
Ev — \ (p is prime) and «(positive integer) ^p. Of the p2 homographies, there are p
sets, each containing p — 1 equivalent (each generates the same p set of points)
homographies. For example, the two equivalent homographies (xh E ^ - 1 ^ , Exz) and
(xi, Ex2, E p _ 1 x 3 ) are found in only one of the p sets. If #2 and Xz are interchanged
and one excludes those collineations which relate to perfect points [W. R. Hutcherson
and J. C. Morelock, Concerning a pattern f or perfect points, Bull. Amer. Math. Soc.
Abstract 60-6-554] it is discovered t h a t there are exactly {p — 1)/2 distinct nonequivalent sets of homographies for each prime number p. (Received June 9, 1955.)
AMS 1955
799/. Harold W. Kuhn: On certain convex polyhedra.
Let Tn be the set of tours (i.e., cyclic arrangements of 1, • • • , n) represented as
n by n permutation matrices t — {Uj). Let Cn denote the convex hull of Tn in n2dimensional Euclidean space. The polyhedron C„ spans the (w2 —3w + 1)-dimensional
linear variety of all x = (Xi3) with xu = 0 and ^jXu = 22*ff»j = 1 for all i and j . Every
nonzero matrix b = (bij) defines a half-space ^bijXij^p
supporting Cn by the rule:
5 5=8minimum of ^bijhjAMERICAN
MATHEMATICAL
[November
(3
for t — (tij)Ç.T
All faces ofSOCIETY
Cn (i.e., (w2 — 3n)-dimensional
n.
intersections of Cn with supporting hyperplanes) can be obtained from non-negative,
integral b. For n<5, matrices of zeros and ones suffice. For n = 5, the following two
classes: x i 2 +#13+^21+#25+#3i+#34+^42+#53^1 (60 distinct faces with renumbering
Of 1, • • • , 5) and Xi3+2Xi 4 +Xi5+^23+^25+^31+X32+^4lH-X42+^46+2X51^2 (120
faces) together with the 210 faces given by Heller [Bull. Amer. Math. Soc. Abstract
59-6-664] describe C& in an irredundant manner. (Received July 13, 1955.)
800/. B. Z. Linfield: Integral and matric geometry.
With functions as coordinates and integrals as distances, any orbit, any velocity
or acceleration of a particle in its orbit, any surface, any space, any direction in a
space, is determined by one scalar equation. Each set of points, be it a curve, a
surface, a space, or a point like a centroid, is characterized by one scalar equation
ƒ = / ( s , u)} where each value of u determines one point in the set, and where the number of components in u determines the dimension of this /-space ƒ (z, u). The square of
Letter from H. Kuhn to G. B. Dantzig
June 10, 1954
“You ask for linear conditions that characterize faces and
supporting planes of the convex hull of tours.”
Letter from H. Kuhn to G. B. Dantzig
Letter from D. Fulkerson
to I. Heller,
March 11, 1954
Acceptance Letter,
G. B. Dantzig, August 25,
1954
Referee’s
Report
“If it is true (and this referee doubts that it is true)
that restraints other than loop conditions must be
used (as stated on p. 10) then a proof of this fact
should be included.”
Ed Paxson,
15-City TSP
W. L. Easton, 1958
Branch-and-Bound Tree,
Easton, 1958
OR 7 (1959)
OR 7 (1959)
“. . . judging from the number of queries we have
received from readers, this method was not
OR 7 (1959)
elaborated sufficiently to make the proposal clear.”
giving complete proofs of results of exceptional interest are also solicited.
OUTLINE OF AN ALGORITHM FOR INTEGER
SOLUTIONS TO LINEAR PROGRAMS
BY RALPH E. GOMORY1
Communicated by A. W. Tucker, May 3, 1958
Bull. AMS 64
(1958)
The problem of obtaining the best integer solution to a linear program comes up in several contexts. The connection with combinatorial problems is given by Dantzig in [ l ] , the connection with problems involving economies of scale is given by Markowitz and Manne
[3 ] in a paper which also contains an interesting example of the effect
of discrete variables on a scheduling problem. Also Dreyfus [4] has
discussed the role played by the requirement of discreteness of variables in limiting the range of problems amenable to linear programming techniques.
It is the purpose of this note to outline a finite algorithm for obtaining integer solutions to linear programs. The algorithm has been
programmed successfully on an E101 computer and used to run off
the integer solution to small (seven or less variables) linear programs
completely automatically.
The algorithm closely resembles the procedures already used by
Dantzig, Fulkerson and Johnson [2], and Markowitz and Manne [3]
to obtain solutions to discrete variable programming problems. Their
procedure is essentially this. Given the linear program, first maximize
the objective function using the simplex method, then examine the
solution. If the solution is not in integers, ingenuity is used to formulate a new constraint that can be shown to be satisfied by the still
unknown integer solution but not by the noninteger solution already
attained. This additional constraint is added to the original ones, the
solution already attained becomes nonfeasible, and a new maximum
satisfying the new constraint is sought. This process is repeated until
an integer maximum is obtained, or until some argument shows that
a nearby integer point is optimal. What has been needed to transform
this procedure into an algorithm is a systematic method for generating
1
This work has been supported in part by the Princeton-IBM Mathematics Research Project.
275
giving complete proofs of results of exceptional interest are also solicited.
OUTLINE OF AN ALGORITHM FOR INTEGER
SOLUTIONS TO LINEAR PROGRAMS
BY RALPH E. GOMORY1
Communicated by A. W. Tucker, May 3, 1958
Bull. AMS 64
(1958)
The problem of obtaining the best integer solution to a linear program comes up in several contexts. The connection with combinatorial problems is given by Dantzig in [ l ] , the connection with problems involving economies of scale is given by Markowitz and Manne
[3 ] in a paper which also contains an interesting example of the effect
of discrete variables on a scheduling problem. Also Dreyfus [4] has
discussed the role played by the requirement of discreteness of variables in limiting the range of problems amenable to linear programming techniques.
It is the purpose of this note to outline a finite algorithm for obtaining integer solutions to linear programs. The algorithm has been
programmed successfully on an E101 computer and used to run off
the integer solution to small (seven or less variables) linear programs
completely automatically.
The algorithm closely resembles the procedures already used by
Dantzig, Fulkerson and Johnson [2], and Markowitz and Manne [3]
to obtain solutions to discrete variable programming problems. Their
procedure is essentially this. Given the linear program, first maximize
the objective function using the simplex method, then examine the
solution. If the solution is not in integers, ingenuity is used to formulate a new constraint that can be shown to be satisfied by the still
unknown integer solution but not by the noninteger solution already
attained. This additional constraint is added to the original ones, the
solution already attained becomes nonfeasible, and a new maximum
satisfying the new constraint is sought. This process is repeated until
an integer maximum is obtained, or until some argument shows that
a nearby integer point is optimal. What has been needed to transform
this procedure into an algorithm is a systematic method for generating
1
This work has been supported in part by the Princeton-IBM Mathematics Research Project.
275
giving complete proofs of results of exceptional interest are also solicited.
OUTLINE OF AN ALGORITHM FOR INTEGER
SOLUTIONS TO LINEAR PROGRAMS
BY RALPH E. GOMORY1
Communicated by A. W. Tucker, May 3, 1958
The problem of obtaining the best integer solution to a linear program comes up in several contexts. The connection with combinatorial problems is given by Dantzig in [ l ] , the connection with problems involving economies of scale is given by Markowitz and Manne
[3 ] in a paper which also contains an interesting example of the effect
of discrete variables on a scheduling problem. Also Dreyfus [4] has
discussed the role played by the requirement of discreteness of variables in limiting the range of problems amenable to linear programming techniques.
It is the purpose of this note to outline a finite algorithm for obtaining integer solutions to linear programs. The algorithm has been
programmed successfully on an E101 computer and used to run off
the integer solution to small (seven or less variables) linear programs
completely automatically.
The algorithm closely resembles the procedures already used by
Dantzig, Fulkerson and Johnson [2], and Markowitz and Manne [3]
to obtain solutions to discrete variable programming problems. Their
procedure is essentially this. Given the linear program, first maximize
the objective function using the simplex method, then examine the
solution. If the solution is not in integers, ingenuity is used to formulate a new constraint that can be shown to be satisfied by the still
unknown integer solution but not by the noninteger solution already
attained. This additional constraint is added to the original ones, the
solution already attained becomes nonfeasible, and a new maximum
satisfying the new constraint is sought. This process is repeated until
an integer maximum is obtained, or until some argument shows that
a nearby integer point is optimal. What has been needed to transform
this procedure into an algorithm is a systematic method for generating
Bull. AMS 64 “The algorithm closely resembles the procedure
already used by Dantzig, Fulkerson and Johnson . . .”
(1958)
1
This work has been supported in part by the Princeton-IBM Mathematics Research Project.
275
History of
Mathematical
Programming
(1991)
History
“During
of
these weeks I learned that others had
Mathematical
thought about the problem and that George Dantzig
Programming
had worked on the traveling salesman problem and
(1991)
had applied special handmade cuts to that.”
Edmonds, 1963
March 16-18, 1964
J. Edmonds:
“For the traveling salesman problem, the vertices of the
associated polyhedron have a simple characterization
despite their number---so might the bounding inequalities
have a simple characterization despite their number. At least
we should hope they have, because finding a really good
traveling salesman algorithm is undoubtedly equivalent to
finding such a characterization.”
R. E. Gomory
“So the number of faces is not the problem. The question is
whether we can get them. And the trouble with the
traveling salesman problem is that we have not, up to now (I
still think it can be done), been able to produce enough of
them easily enough ...”
Gardiner L. Tucker
IBM Director of Research
1963-1969
Mathematical Programming 1 (1971) 6-25. North-Holland Publishing Company
THE TRAVELING-SALESMAN PROBLEM
AND MINIMUM SPANNING TREES: PART 1I *
Michael HELD
IBM Systems Research Institute New York, New York, U.S.A.
and
Richard M. KARP **
University of California, Berkeley, California, U.S.A.
Received 19 October 1970
Held-Karp
Math Prog 1
(1971)
The relationship between the symmetric traveling-salesman problem and the minimum
spanning tree problem yields a sharp lower bound on the cost of an optimum tour. An efficient iterative method for approximating this bound closely from below is presented. A
branch-and-bound procedure based upon these considerations has easily produced proven
optimum solutions to all traveling-salesman problems presented to it, ranging in size up to
sixty-four cities. The bounds used are so sharp that the search trees are minuscule compared to
those normally encountered in combinatorial problems of this type.
0. Introduction
In a previous paper [7], the authors explored the relationship between the symmetric traveling-salesman problem and the minimum
spanning tree problem. By means of this relationship a lower bound on
the cost of an optimum tour was derived which is quite sharp, and hence
of value in connection with branch-and-bound procedures. The methods
proposed in [7] for computing this bound proved inadequate. The
present paper gives an efficient method for approximating the bound
closely from below, and reports on the use of this method in a highly
successful algorithm for the e x a c t solution of symmetric travelingsalesman problems.
* This paper was presented at the 7th Mathematical Programming Symposium 1970, The
Hague, The Netherlands.
** This research has been partially supported by the National Science Foundation under
Grant GP-25081 with the University of California. Reproduction in whole or in part is
permitted for any purpose of the United States Government.
TURINGA WARDLECTURE
COMBINATORICS, COMPLEXITY,
AND RANDOMNESS
The 1985 Turing Award winner presents his perspective on the development
of the field that has come to be called theoretical computer science.
RICHARD M. KARP
This lecture is dedicated to the memory
Abraham Louis Karp.
of my father,
I am honored and pleased to be the recipient of this
year’s Turing Award. As satisfying as it is to receive
such recognition, I find that my greatest satisfaction
as a researcher has stemmed from doing the research
itself. and from the friendships I have formed along the
way. I would like to roam with you through my 25
years as a researcher in the field of combinatorial
algorithms and computational
complexity, and tell you
about some of the concepts that have seemed important
to me, and about some of the people who have inspired
and influenced me.
Turing Award
Lecture
BEGINNINGS
My entry into the computer field was rather accidental.
Having graduated from Harvard College in 1955 with a
degree in mathematics, I was c:onfronted with a decision as to what to do next. Working for a living had
little appeal, so graduate school was the obvious choice.
One possibility was to pursue a career in mathematics,
but the field was thl?n in the heyday of its emphasis on
abstraction and generality, and the concrete and applicable mathematics that I enjoyed the most seemed to
be out of fashion.
And so, almost by default, I entered the Ph.D. program at the Harvard Computation Laboratory. Most of
the topics that were to become the bread and butter of
the computer scienc:e curriculum had not even been
thought of then, and so I took an eclectic collection of
courses: switching theory, numerical analysis, applied
mathematics, probability and statistics, operations research, electronics, and mathematical linguistics. While
the curriculum left much to be desired in depth and
01986 ACM 0001.0762/66,'0200-0096
98
Communications
of the ,4CM
75a
coherence, there was a very special spirit in the air; we
knew that we were witnessing the birth of a new scientific discipline centered on the computer. I discovered
that I found beauty and elegance in the structure of
algorithms, and that I had a knack for the discrete
mathematics that formed the basis for the study of
computers and computation. In short, I had stumbled
more or less by accident into a field that was very
much to my liking.
EASY AND HARD COMBINATORIAL
PROBLEMS
Ever since those early days, I have had a special interest in combinatorial search problems-problems
that
can be likened to jigsaw puzzles where one has to assemble the parts of a structure in a particular way.
Such problems involve searching through a finite, but
extremely large, structured set of possible solutions,
patterns, or arrangements, in order to find one that
satisfies a stated set of conditions. Some examples of
such problems are the placement and interconnection
of components on an integrated circuit chip, the scheduling of the National Football League, and the routing
of a fleet of school buses.
Within any one of these combinatorial
puzzles lurks
the possibility of a combinatorial
explosion. Because of
the vast, furiously growing number of possibilities that
have to be searched through, a massive amount of computation may be encountered unless some subtlety is
used in searching through the space of possible solutions. I’d like to begin the technical part of this talk by
telling you about some of my first encounters with
combinatorial explosions.
My first defeat at the hands of this phenomenon
came soon after I joined the IBM Yorktown Heights
Research Center in 1959. I was assigned to a group
headed by J. P. Roth, a distinguished algebraic topolo-
February 1986
Volume 29 Number 2
TURINGA WARDLECTURE
COMBINATORICS, COMPLEXITY,
AND RANDOMNESS
The 1985 Turing Award winner presents his perspective on the development
of the field that has come to be called theoretical computer science.
“After a long series of unsuccessful experiments, Held and I
stumbled upon a powerful method of computing lower
bounds. This bounding technique allowed us to prune the
search
severely, so that we were able to solve problems
Turing
Award
with as many as 65 cities. I don’t think any of my
1985
theoretical results have provided as great a thrill as the
sight of the numbers pouring out of the computer on the
night Held and I first tested our bounding method.”
RICHARD M. KARP
This lecture is dedicated to the memory
Abraham Louis Karp.
of my father,
I am honored and pleased to be the recipient of this
year’s Turing Award. As satisfying as it is to receive
such recognition, I find that my greatest satisfaction
as a researcher has stemmed from doing the research
itself. and from the friendships I have formed along the
way. I would like to roam with you through my 25
years as a researcher in the field of combinatorial
algorithms and computational
complexity, and tell you
about some of the concepts that have seemed important
to me, and about some of the people who have inspired
and influenced me.
BEGINNINGS
My entry into the computer field was rather accidental.
Having graduated from Harvard College in 1955 with a
degree in mathematics, I was c:onfronted with a decision as to what to do next. Working for a living had
little appeal, so graduate school was the obvious choice.
One possibility was to pursue a career in mathematics,
but the field was thl?n in the heyday of its emphasis on
abstraction and generality, and the concrete and applicable mathematics that I enjoyed the most seemed to
be out of fashion.
And so, almost by default, I entered the Ph.D. program at the Harvard Computation Laboratory. Most of
the topics that were to become the bread and butter of
the computer scienc:e curriculum had not even been
thought of then, and so I took an eclectic collection of
courses: switching theory, numerical analysis, applied
mathematics, probability and statistics, operations research, electronics, and mathematical linguistics. While
the curriculum left much to be desired in depth and
01986 ACM 0001.0762/66,'0200-0096
98
Communications
of the ,4CM
75a
coherence, there was a very special spirit in the air; we
knew that we were witnessing the birth of a new scientific discipline centered on the computer. I discovered
that I found beauty and elegance in the structure of
algorithms, and that I had a knack for the discrete
mathematics that formed the basis for the study of
computers and computation. In short, I had stumbled
more or less by accident into a field that was very
much to my liking.
EASY AND HARD COMBINATORIAL
PROBLEMS
Ever since those early days, I have had a special interest in combinatorial search problems-problems
that
can be likened to jigsaw puzzles where one has to assemble the parts of a structure in a particular way.
Such problems involve searching through a finite, but
extremely large, structured set of possible solutions,
patterns, or arrangements, in order to find one that
satisfies a stated set of conditions. Some examples of
such problems are the placement and interconnection
of components on an integrated circuit chip, the scheduling of the National Football League, and the routing
of a fleet of school buses.
Within any one of these combinatorial
puzzles lurks
the possibility of a combinatorial
explosion. Because of
the vast, furiously growing number of possibilities that
have to be searched through, a massive amount of computation may be encountered unless some subtlety is
used in searching through the space of possible solutions. I’d like to begin the technical part of this talk by
telling you about some of my first encounters with
combinatorial explosions.
My first defeat at the hands of this phenomenon
came soon after I joined the IBM Yorktown Heights
Research Center in 1959. I was assigned to a group
headed by J. P. Roth, a distinguished algebraic topolo-
February 1986
Volume 29 Number 2
S. Hong, Thesis, 1972
TSP Polytope: Groetschel and Padberg
Math Prog 8 (1975), 378-381
ZOR 21 (1977), 33-64
Math Prog 16 (1979), 265-280
Math Prog 16 (1979), 281-302
Thesis, Groetschel, 1977
Chvatal Combs: Math Prog 5 (1973), 29-40
M. Groetschel,
June, 1976
LARGE-SCALE SYMMETRIC TRAVELLING SALESMAN PROBLEMS
Crowder-Padberg,
Man Sci 26 (1980)
505
routine zero-one solutions were found that constituted collections of subtours in the
respective graph having the same objective function value as the optimal tour, which
was found in the respective last application of the MIP/370 routine.
LIN318A and LIN318B are both runs of the 318-city problem due to Lin and
Kemigham [12]. The first run LIN318A was made with the output from the cuttingplane method (TSP) using the suboptimal tour of length 41,349 found in [16]. After
three applications of the MIP/370 routine the program halted and the optimal tour of
length 41,345 was found. The optimal tour is displayed in Figure 2. Using Stirling's
approximation formula for n!, the tour displayed in Figure 2 is the (possibly unique)
tour (having one arc fixed) from among 10**' tours that are possible among 318 points
and have one arc fixed. Assuming that one could possibly enumerate lO' solutions
(tours) per second on a computer it would thus take roughly 10*^' years of computing
to establish the optimality of this tour by exhaustive enumeration. Solving the TSP
output to optimality took less than 6 minutes of CPU-rime. The run LIN318B was
essentially made to validate the previous run; in a way, it was sort of a check of
internal consistency. First, the TSP package (the cutting-plane procedure described in
Section 1) was re-run with the optimal tour as the starting solution. The results of this
run are displayed in Table I in the row labelled LIN318B. As a result of the (slightly)
smaller gap, the TSP-program fixed more variables than previously and the problem
generated as input for the MPSX-MIP/370 routine had 495 rows and 1,144 variables.
313.0 .
63.0
-500.0
500.0
0.
1500.4
2500.4
3MX).4
4500.4
1000.0
^)00.4
3000.4
«X».4
FIGURE 2.
An Optimal Ordering
1. Pilsner Urquell
2. Koenig Pilsener
3. Guenzburger Edel-Pils
4. Warsteiner
5. Paulaner Pils
6. Budweiser
7. Riegele Herren-Pils
8. Beck’s Bier
9. Jever Pilsener
10. Fuerstenberg Pilsener
Groetschel, Juenger, Reinelt
OR 32 (1984), 1195-1220
Math Prog 33 (1985), 28-42
Math Prog 33 (1985), 43-60
Branch-and-Cut
Algorithm
OR 32 (1984)
Groetschel-Holland, 1987
Padberg-Rinaldi, 1988
Padberg-Rinaldi,
1988
Groetschel, Lovasz, Schrijver
(1988)
From the Preface:
“The central result proved and applied in this book is,
roughly, the following. If K is a convex set, and if we
Groetschel, Lovasz, Schrijver
can decide
in polynomial time whether a given vector
(1988)
belongs to K, then we can optimize any linear function
over K in polynomial time.”
Open Problems
and Conjectures
$$$
G. Cornuejols, 2001
Annals of Mathematics, 164 (2006), 51–229
The strong perfect graph theorem
By Maria Chudnovsky, Neil Robertson,∗ Paul Seymour,∗*
and Robin Thomas∗∗∗
Abstract
Chudnovsky, Robertson,
Seymour, Thomas
Ann. Math 164
(2006)
A graph G is perfect if for every induced subgraph H, the chromatic
number of H equals the size of the largest complete subgraph of H, and G is
Berge if no induced subgraph of G is an odd cycle of length at least five or the
complement of one.
The “strong perfect graph conjecture” (Berge, 1961) asserts that a graph
is perfect if and only if it is Berge. A stronger conjecture was made recently by
Conforti, Cornuéjols and Vušković — that every Berge graph either falls into
one of a few basic classes, or admits one of a few kinds of separation (designed
so that a minimum counterexample to Berge’s conjecture cannot have either
of these properties).
In this paper we prove both of these conjectures.
1. Introduction
We begin with definitions of some of our terms which may be nonstandard.
All graphs in this paper are finite and simple. The complement G of a graph
G has the same vertex set as G, and distinct vertices u, v are adjacent in G
just when they are not adjacent in G. A hole of G is an induced subgraph of G
which is a cycle of length at least 4. An antihole of G is an induced subgraph
of G whose complement is a hole in G. A graph G is Berge if every hole and
antihole of G has even length.
A clique in G is a subset X of V (G) such that every two members of
X are adjacent. A graph G is perfect if for every induced subgraph H of G,
*Supported by ONR grant N00014-01-1-0608, NSF grant DMS-0071096, and AIM.
∗∗ Supported by ONR grants N00014-97-1-0512 and N00014-01-1-0608, and NSF grant
DMS-0070912.
∗∗∗ Supported by ONR grant N00014-01-1-0608, NSF grants DMS-9970514 and DMS0200595, and AIM.
Combinatorial
Optimization: Polyhedra
and Efficiency
A. Schrijver, 2003
From the Preface:
Combinatorial
Optimization: Polyhedra
“Pioneered
and Efficiency by the work of Jack Edmonds,
polyhedral combinatorics has proved to be a
A.most
Schrijver,
2003 coherent, and unifying tool
powerful,
throughout combinatorial optimization.”