Fifty-plus years of combinatorial integer programming

50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art

Volumn , Issue , 2010, Pages 387-430

  Cook, William ^a  

^a Georgia Institute of Technology   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 81155160480     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1007/978-3-540-68279-0_12     Document Type: Chapter

References (144)

1. F. Alizadeh, Interior point methods in semidefinite programming with applications to combinatorial optimization, SIAM Journal on Optimization 5 (1995) 13-51.
2. American Mathematical Society, The Sixth Symposium in Applied Mathematics, Bulletin of the American Mathematical Society 59 (1953) 513-514.
3. American Mathematical Society, The Summer Meeting in Kingston, Bulletin of the American Mathematical Society 59 (1953) 515-568.
4. E. Balas and M.W. Padberg, On the set-covering problem, Operations Research 20 (1972) 1152-1161.
5. E. Balas and M.W. Padberg, On the set-covering problem: II. An algorithm for set partitioning, Operations Research 23 (1975) 74-90.
6. R. Bellman, An introduction to the theory of dynamic programming, Research Report R-245, RAND Corporation, Santa Monica, California, USA, 1953.
7. R. Bellman, Dynamic Programming, Princeton University Press, Princeton, New Jersey, USA, 1957.
8. R. Bellman, Combinatorial processes and dynamic programming, Combinatorial Analysis (R. Bellman and M. Hall, Jr., eds.), American Mathematical Society, Providence, Rhode Island, USA, 1960, pp. 217-249.
9. R. Bellman, Dynamic programming treatment of the travelling salesman problem, Journal of the Association for Computing Machinery 9 (1962) 61-63.
10. C. Berge, F ärbung von Graphen, deren sämtliche bzw. deren ungerade Kreise starr sind (Zusammenfassung), Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961) 114-115.
11. C. Berge, Sur certains hypergraphes géné ralisant les graphes bipartis, Combinatorial Theory and its Applications I (P. Erdos, A. Renyi, and V. Sós, eds.), Colloq. Math. Soc. János Bolyai 4 (1970) 119-133.
12. C. Berge, Balanced matrices, Mathematical Programming 2 (1972) 19-31.
13. R.E. Bixby, Solving real-world linear programs: A decade and more of progress, Operations Research 50 (2002) 3-15.
14. F. Bock, An algorithm for solving "traveling-salesman," and related network optimization problems, Research Report, Armour Research Foundation, Presented at the Operations Research Society of America Fourteenth National Meeting, St. Louis, October 24, 1958.
15. M. Chudnovsky, G. Cornuéjols, X. Liu, P.D. Seymour, and K. Vušković, Recognizing Berge graphs, Combinatorica 25 (2005) 143-186.
16. M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem, Annals of Mathematics 164 (2006) 51-229.
17. V. Chvátal, Edmonds polytopes and a hierarchy of combinatorial problems, Discrete Mathematics 4 (1973) 305-337.
18. V. Chvátal, Edmonds polytopes and weakly hamiltonian graphs, Mathematical Programming 5 (1973) 29-40.
19. V. Chvátal, On certain polyhedra associated with graphs, Journal of Combinatorial Theory B 18 (1975) 138-154.
20. V. Chvátal, Cutting planes in combinatorics, European Journal of Combinatorics 6 (1985) 217-226.
21. V. Chvátal, W. Cook, and M. Hartmann, On cutting-plane proofs in combinatorial optimization, Linear Algebra and Its Applications 114/115 (1989) 455-499.
22. M. Conforti, G. Cornuéjols, and M.R. Rao, Decomposition of balanced matrices, Journal of Combinatorial Theory, Series B 77 (1999) 292-406.
23. G. Cornuéjols, Combinatorial Optimization: Packing and Covering, SIAM, Philadelphia, USA, 2001.
24. G.A. Croes, A method for solving traveling-salesman problems, Operations Research 6 (1958) 791-812.
25. H. Crowder and M.W. Padberg, Solving large-scale symmetric travelling salesman problems to optimality, Management Science 26 (1980) 495-509.
26. W.H. Cunningham and A.B. Marsh, III, A primal algorithm for optimum matching, Mathematical Programming Study 8 (1978) 50-72.
27. G.B. Dantzig, Discrete-variable extremum problems, Operations Research 5 (1957) 266-277.
28. G.B. Dantzig, D.R. Fulkerson, and S.M. Johnson, Solution of a large scale traveling salesman problem, Technical Report P-510, RAND Corporation, Santa Monica, California, USA, 1954.
29. G.B. Dantzig, D.R. Fulkerson, and S.M. Johnson, Solution of a large-scale traveling-salesman problem, Operations Research 2 (1954) 393-410.
30. G.B. Dantzig, D.R. Fulkerson, and S.M. Johnson, On a linear-programming, combinatorial approach to the traveling-salesman problem, Operations Research 7 (1959) 58-66.
31. G.B. Dantzig and A.J. Hoffman, Dilworth's theorem on partially ordered sets, Linear Inequalities and Related Systems (H.W. Kuhn and A.W. Tucker, eds.), Princeton University Press, Princeton, New Jersey, USA, 1956, pp. 207-214.
32. W.L. Eastman, Linear Programming with Pattern Constraints, Ph.D. Thesis, Department of Economics, Harvard University, Cambridge, Massachusetts, USA, 1958.
33. J. Edmonds, Paths, trees, and flowers, Working paper, National Bureau of Standards and Princeton University, February 1963.
34. J. Edmonds, Maximum matching and a polyhedron with 0,1-vertices, Journal of Research of the National Bureau of Standards 69B (1965) 125-130.
35. J. Edmonds, Paths, trees, and flowers, Canadian Journal of Mathematics 17 (1965) 449-467.
36. J. Edmonds, The Chinese postman's problem, Bulletin of the Operations Research Society of America 13 (1965) B-73.
37. J. Edmonds, Optimum branchings, Journal of Research National Bureau of Standards 71B (1967) 233-240.
38. J. Edmonds, Submodular functions, matroids, and certain polyhedra, Combinatorial Structures and Their Applications (R. Guy, H. Hanani, N. Sauer, and J. Schönheim, eds.), Gordon and Breach, New York, USA, 1970, pp. 69-87.
39. J. Edmonds, Matroids and the greedy algorithm, Mathematical Programming 1 (1971) 127-136.
40. J. Edmonds, Matroid intersection, Discrete Optimization I (P.L. Hammer, E.L. Johnson, and B.H. Korte, eds.), North-Holland, 1979, pp. 39-49.
41. J. Edmonds and R. Giles, A min-max relation for submodular functions on a graph, Studies in Integer Programming (P.L. Hammer, E.L. Johnson, B.H. Korte, and G.L. Nemhauser, eds.), Annals of Discrete Mathematics 1 (1977) 185-204.
42. J. Edmonds and E.L. Johnson, Matching, a well-solved class of integer linear programs, Combinatorial Structures and Their Applications (R. Guy, H. Hanani, N. Sauer, and J. Schönheim, eds.), Gordon and Breach, New York, USA, 1970, pp. 89-92.
43. J. Edmonds and E.L. Johnson, Matching, Euler tours, and the Chinese postman, Mathematical Programming 5 (1973) 88-124.
44. J. Egerváry, Matrixok kombinatorius tulajonságairó l, Matematikai és Fizikai Lapok 38 (1931) 16-28.
45. M.M. Flood, Merrill Flood (with Albert Tucker), Interview of Merrill Flood in San Francisco on 14 May 1984, The Princeton Mathematics Community in the 1930s, Transcript Number 11 (PMC11), Princeton University. (1984).
46. L.R. Ford, Jr. and D.R. Fulkerson, Maximal flow through a network, Canadian Journal of Mathematics 8 (1956) 399-404.
47. A. Frank, Kernel systems of directed graphs, Acta Scientiarum Mathematicarum [Szeged] 41 (1979) 63-76.
48. D.R. Fulkerson, Blocking and anti-blocking pairs of polyhedra, Mathematical Programming 1 (1971) 168-194.
49. D.R. Fulkerson, Anti-blocking polyhedra, Journal of Combinatorial Theory, Series B 12 (1972) 50-71.
50. D.R. Fulkerson, A.J. Hoffman, and R. Oppenheim, On balanced matrices, Mathematical Programming Study 1 (1974) 120-132.
51. D. Gale, A theorem on flows in networks, Pacific Journal of Mathematics 7 (1957) 1073-1082.
52. F.R. Giles and W.R. Pulleyblank, Total dual integrality and integer polyhedra, Linear Algebra and Its Applications 25 (1979) 191-196.
53. M. Goemans and D. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems, Journal of the Association of Computing Machinery 42 (1995) 1115-1145.
54. R.E. Gomory, Outline of an algorithm for integer solutions to linear programs, Bulletin of the American Mathematical Society 64 (1958) 275-278.
55. R.E. Gomory, The traveling salesman problem, Proceedings of the IBM Scientific Computing Symposium on Combinatorial Problems, IBM, White Plains, New York, USA, 1966, pp. 93-121.
56. R.E. Gomory, Early integer programming, History of Mathematical Programming-A Collection of Personal Reminiscences (J.K. Lenstra, A.H.G. Rinnooy Kan, and A. Schrijver, eds.), North-Holland, 1991, pp. 55-61.
57. R.H. Gonzales, Solution to the traveling salesman problem by dynamic programming on the hypercube, Technical Report Number 18, Operations Research Center, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA, 1962.
58. M. Grötschel, Polyedrische Charakterisierungen kombinatorischer Optimierungsprobleme, Anton Hain Verlag, Meisenheim/Glan, Germany, 1977.
59. M. Grötschel, On the symmetric travelling salesman problem: Solution of a 120-city problem, Mathematical Programming Study 12 (1980) 61-77.
60. M. Grötschel and O. Holland, Solution of large-scale symmetric travelling salesman problems, Mathematical Programming 51 (1991) 141-202.
61. M. Grötschel, M. Jünger, and G. Reinelt, A cutting plane algorithm for the linear ordering problem, Operations Research 32 (1984) 1195-1220.
62. M. Grötschel, M. Jünger, and G. Reinelt, On the acyclic subgraph polytope, Mathematical Programming 33 (1985) 28-42.
63. M. Grötschel, M. Jünger, and G. Reinelt, Facets of the linear ordering polytope, Mathematical Programming 33 (1985) 43-60.
64. M. Grötschel, L. Lovász, and A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1 (1981) 169-197.
65. M. Grötschel, L. Lovász, and A. Schrijver, Polynomial algorithms for perfect graphs, Topics on Perfect Graphs (C. Berge and V. Chvátal, eds.), Annals of Discrete Mathematics 21 (1984) 325-356.
66. M. Grötschel, L. Lovász, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer. (1988).
67. M. Grötschel and G.L. Nemhauser, George Dantzig's contributions to integer programming, Discrete Optimization 5 (2008) 168-173.
68. M. Grötschel and W.R. Pulleyblank, Clique tree inequalities and the symmetric travelling salesman problem, Mathematics of Operations Research 11 (1986) 537-569.
69. M. Held and R.M. Karp, A dynamic programming approach to sequencing problems, Journal of the Society of Industrial and Applied Mathematics 10 (1962) 196-210.
70. M. Held and R.M. Karp, The traveling-salesman problem and minimum spanning trees, Operations Research 18 (1970) 1138-1162.
71. M. Held and R.M. Karp, The traveling-salesman problem and minimum spanning trees: Part II, Mathematical Programming 1 (1971) 6-25.
72. I. Heller, On the problem of the shortest path between points, I. Abstract 664t, Bulletin of the American Mathematical Society 59 (1953) 551.
73. C. Helmberg, Semidefinite Programming for Combinatorial Optimization, Habilitationsschrift, TU Berlin, ZIB-Report ZR-00-34, Konrad-Zuse-Zentrum Berlin, 2000.
74. A.J. Hoffman, Generalization of a theorem of König, Journal of the Washington Academy of Sciences 46 (1956) 211-212.
75. A.J. Hoffman, Linear programming, Applied Mechanics Reviews 9 (1956) 185-187.
76. A.J. Hoffman, A generalization of max flow-min cut, Mathematical Programming 6 (1974) 352-359.
77. A.J. Hoffman and J.B. Kruskal, Integral boundary points of convex polyhedra, Linear Inequalities and Related Systems (H.W. Kuhn and A.W. Tucker, eds.), Princeton University Press, Princeton, New Jersey, USA, 1956, pp. 223-246.
78. A.J. Hoffman and H.W. Kuhn, Systems of distinct representatives and linear programming, The American Mathematical Monthly 63 (1956) 455-460.
79. A.J. Hoffman and R. Oppenheim, Local unimodularity in the matching polytope, Algorithmic Aspects of Combinatorics (B.R. Alspach, P. Hell, and D.J. Miller, eds.), North-Holland, 1978, pp. 201-209.
80. A.J. Hoffman and D.E. Schwartz, On lattice polyhedra, Combinatorics Volume I (A. Hajnal and V.T. Sós, eds.), North-Holland, 1978, pp. 593-598.
81. A.J. Hoffman and P. Wolfe, History, The Traveling Salesman Problem (E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, and D.B. Shmoys, eds.), John Wiley & Sons, Chichester, UK, 1985, pp. 1-15.
82. O.A. Holland, Schnittebenenverfahren für Travelling-Salesman und verwandte Probleme, Ph.D. Thesis, Universität Bonn, Bonn, Germany, 1987.
83. S. Hong, A Linear Programming Approach for the Traveling Salesman Problem, Ph.D. Thesis, Johns Hopkins University, Baltimore, Maryland, USA, 1972.
84. M. Jünger, Polyhedral Combinatorics and the Acyclic Subdigraph Problem, Heldermann Verlag, Berlin, Germany, 1985.
85. M. Jünger, G. Reinelt, and S. Thienel, Practical problem solving with cutting plane algorithms, Combinatorial Optimization (W. Cook, L. Lovász, and P. Seymour, eds.), DIMACS Series in Discrete Mathematics and Theoretical Computer Science 20, American Mathematical Society, Providence, Rhode Island, USA, 1995, pp. 111-152.
86. R.M. Karp, Combinatorics, complexity, and randomness, Communications of the ACM 29 (1986) 98-109.
87. R.M. Karp and C.H. Papadimitriou, On linear characterization of combinatorial optimization problems, SIAM Journal on Computing 11 (1982) 620-632.
88. L.G. Khachiyan, A polynomial algorithm in linear programming, Soviet Mathematics Doklady 20 (1979) 191-194.
89. B. Korte and W. Oberhofer, Zwei Algorithmen zur Lösung eines komplexen Reihenfolgeproblems, Mathematical Methods of Operations Research 12 (1968) 217-231.
90. B. Korte and W. Oberhofer, Zur Triangulation von Input-Output-Matrizen, Jahrbücher für Nationalökonomie und Statistik 182 (1969) 398-433.
91. H.W. Kuhn, On certain convex polyhedra, Abstract 799t, Bulletin of the American Mathematical Society 61 (1955) 557-558.
92. H.W. Kuhn, The Hungarian method for the assignment problem, Naval Research Logistics Quarterly 2 (1955) 83-97.
93. H.W. Kuhn, On the origin of the Hungarian method, History of Mathematical Programming-A Collection of Personal Reminiscences (J.K. Lenstra, A.H.G. Rinnooy Kan, and A. Schrijver, eds.), North-Holland, 1991, pp. 77-81.
94. H.W. Kuhn, 57 years of close encounters with George, George Dantzig Memorial Site, INFORMS, 2008, available at http://www2.informs.org/History/ dantzig/articleskuhn.html.
95. H.W. Kuhn, Email letter sent on December 15, 2008.
96. H.W. Kuhn and A.W. Tucker, eds. Linear Inequalities and Related Systems, Princeton University Press, Princeton, New Jersey, USA, 1956.
97. F. Lambert, The traveling-salesman problem, Cahiers du Centre de Recherche Opérationelle 2 (1960) 180-191.
98. A.H. Land and A.G. Doig, An automatic method of solving discrete programming problems, Econometrica 28 (1960) 497-520.
99. A.H. Land and S. Powell, A survey of the operational use of ILP models, History of Integer Programming: Distinguished Personal Notes and Reminiscences (K. Spielberg and M. Guignard-Spielberg, eds.), Annals of Operations Research 149 (2007) 147-156.
100. M. Laurent and F. Rendl, Semidefinite programming and integer programming, Handbook on Discrete Optimization (K. Aardal, G.L. Nemhauser, and R. Weismantel, eds.), Elsevier, 2005, pp. 393-514.
101. J.K. Lenstra, Sequencing by Enumerative Methods, Mathematical Centre Tracts 69, Mathematisch Centrum, Amsterdam, 1977.
102. A.N. Letchford and A. Lodi, Primal cutting plane algorithms revisited, Mathematical Methods of Operations Research 56 (2002) 67-81.
103. S. Lin and B.W. Kernighan, An effective heuristic algorithm for the traveling-salesman problem, Operations Research 21 (1973) 498-516.
104. J.D. Little, K.G. Murty, D.W. Sweeney, and C. Karel, An algorithm for the traveling salesman problem, Operations Research 11 (1963) 972-989.
105. L. Lovász, A characterization of perfect graphs, Journal of Combinatorial Theory, Series B 13 (1972) 95-98.
106. L. Lovász, Normal hypergraphs and the perfect graph conjecture, Discrete Mathematics 2 (1972) 253-267.
107. L. Lovász, On the Shannon capacity of graphs, IEEE Transactions on Information Theory 25 (1979) 1-7.
108. L. Lovász and A. Schrijver, Cones of matrices and set-functions, and 0-1 optimization, SIAM Journal on Optimization 1 (1991) 166-190.
109. C.L. Lucchesi and D.H. Younger, A minimax theorem for directed graphs, The Journal of the London Mathematical Society 17 (1978) 369-374.
110. A.S. Manne and H.M. Markowitz, On the solution of discrete programming problems, Technical Report P-711, RAND Corporation, Santa Monica, California, USA, 1956.
111. H.M. Markowitz and A.S. Manne, On the solution of discrete programming problems, Econo-metrica 25 (1957) 84-110.
112. G.T. Martin, Solving the traveling salesman problem by integer linear programming, Technical Report, C-E-I-R, New York, USA, 1966.
113. K. Menger, Bericht ü ber ein mathematisches Kolloquium, Monatshefte für Mathematik und Physik 38 (1931) 17-38.
114. C.A. Micchelli, Selected Papers of Alan Hoffman with Commentary, World Scientific Publishing Company, 2003.
115. P. Miliotis, Using cutting planes to solve the symmetric travelling salesman problem, Mathematical Programming 15 (1978) 177-188.
116. C.E. Miller, A.W. Tucker, and R.A. Zemlin, Integer programming formulation of traveling salesman problems, Journal of the Association for Computing Machinery 7 (1960) 326-329.
117. J. Munkres, Algorithms for the assignment and transportation problems, Journal of the Society for Industrial and Applied Mathematics 5 (1957) 32-33.
118. G.L. Nemhauser and L.E. Trotter, Jr., Properties of vertex packing and independence system polyhedra, Mathematical Programming 6 (1974) 48-61.
119. G.L. Nemhauser and L.E. Trotter, Jr., Vertex packings: Structural properties and algorithms, Mathematical Programming 8 (1975) 232-248.
120. Y. Nesterov and A. Nemirovski, Conic formulation of a convex programming problem and duality, Optimization Methods and Software 1 (1992) 95-115.
121. Y. Nesterov and A. Nemirovski, Interior Point Polynomial Algorithms in Convex Programming, SIAM, Philadelphia, USA, 1994.
122. M.W. Padberg, On the facial structure of set packing polyhedra, Mathematical Programming 5 (1973) 199-215.
123. M.W. Padberg, Mixed-integer programming - 1968 and thereafter, History of Integer Programming: Distinguished Personal Notes and Reminiscences (K. Spielberg and M. Guignard-Spielberg, eds.), Annals of Operations Research 149 (2007) 147-156.
124. M.W. Padberg and S. Hong, On the symmetric travelling salesman problem: A computational study, Mathematical Programming Study 12 (1980) 78-107.
125. M.W. Padberg and M.R. Rao, The Russian method for linear programming III: Bounded integer programming, Research Report 81-39, New York University, Graduate School of Business Administration, 1981.
126. M.W. Padberg and G. Rinaldi, Optimization of a 532-city symmetric traveling salesman problem by branch and cut, Operations Research Letters 6 (1987) 1-7.
127. M.W. Padberg and G. Rinaldi, Facet identification for the symmetric traveling salesman poly-tope, Mathematical Programming 47 (1990) 219-257.
128. M.W. Padberg and G. Rinaldi, A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems, SIAM Review 33 (1991) 60-100.
129. W.R. Pulleyblank, Faces of Matching Polyhedra, Ph.D. Thesis, Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada, 1973.
130. W.R. Pulleyblank and J. Edmonds, Facets of 1-matching polyhedra, Hypergraph Seminar (C. Berge and D. Ray-Chaudhuri, eds.), Springer, 1974, pp. 214-242.
131. G. Reinelt, The Linear Ordering Problem: Algorithms and Applications, Heldermann Verlag, Berlin, Germany, 1985.
132. J. Robinson, On the Hamiltonian game (a traveling salesman problem), Research Memorandum RM-303, RAND Corporation, Santa Monica, California, USA, 1949.
133. A. Schrijver, On cutting planes, Combinatorics 79 Part II (M. Deza and I.G. Rosenberg, eds.), Annals of Discrete Mathematics 9, North-Holland, 1980, pp. 291-296.
134. A. Schrijver, On total dual integrality, Linear Algebra and Its Applications 38 (1981) 27-32.
135. A. Schrijver, Combinatorial Optimization: Polyhedra and Efficiency, Springer, Berlin, Germany, 2003.
136. A. Schrijver, On the history of combinatorial optimization (till 1960), Handbook of Discrete Optimization (K. Aardal, G.L. Nemhauser, and R. Weismantel, eds.), Elsevier, 2005, pp. 1-68.
137. A. Schrijver and P.D. Seymour, A proof of total dual integrality of matching polyhedra, Report ZN 79/77, Stichting Mathematisch Centrum, Amsterdam, 1977.
138. P.D. Seymour, The matroids with the max-flow min-cut property, Journal of Combinatorial Theory, Series B 23 (1977) 289-295.
139. P.D. Seymour, Decomposition of regular matroids, Journal of Combinatorial Theory, Series B 28 (1980) 305-359.
140. P.D. Seymour, How the proof of the strong perfect graph conjecture was found, Gazette des Mathematiciens 109 (2006) 69-83.
141. M. Todd, Semidefinite optimization, Acta Numerica 10 (2001) 515-560.
142. H. Wolkowicz, Semidefinite and cone programming bibliography/comments/ abstracts, 2008, see http://orion.uwaterloo.ca/hwolkowi/henry/book/fronthandbk. d/handbooksdp.html.
143. L.A. Wolsey, Further facet generating procedures for vertex packing polytopes, Mathematical Programming 11 (1976) 158-163.
144. D.B. Yudin and A S. Nemirovski, Evaluation of the informational complexity of mathematical programming problems, Ékonomica i Matematicheskie Metody 12 (1976) 128-142.