Advertisement

Perfect Formulations

  • Michele Conforti
  • Gérard Cornuéjols
  • Giacomo Zambelli
Chapter
Part of the Graduate Texts in Mathematics book series (GTM, volume 271)

Abstract

A perfect formulation of a set \(S \subseteq \mathbb{R}^{n}\) is a linear system of inequalities Ax ≤ b such that \(\mathrm{conv}(S) =\{ x \in \mathbb{R}^{n}\,:\, Ax \leq b\}\). For example, Proposition 1.5 gives a perfect formulation of a 2-variable mixed integer linear set. When a perfect formulation is available for a mixed integer linear set, the corresponding integer program can be solved as a linear program. In this chapter, we present several classes of integer programming problems for which a perfect formulation is known. For pure integer linear sets, a classical case is when the constraint matrix is totally unimodular. Important combinatorial problems on directed or undirected graphs such as network flows and matchings in bipartite graphs have a totally unimodular constraint matrix.

Bibliography

  1. [15]
    A. Atamtürk, Strong formulations of robust mixed 0–1 programming. Math. Program. 108, 235–250 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  2. [17]
    R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network Flows (Prentice Hall, New Jersey, 1993)zbMATHGoogle Scholar
  3. [24]
    E. Balas, Disjunctive programming: properties of the convex hull of feasible points, GSIA Management Science Research Report MSRR 348, Carnegie Mellon University (1974); Published as invited paper in Discrete Appl. Math. 89, 1–44 (1998)Google Scholar
  4. [26]
    E. Balas, Disjunctive programming and a hierarchy of relaxations for discrete optimization problems. SIAM J. Algebr. Discrete Methods 6, 466–486 (1985)zbMATHMathSciNetCrossRefGoogle Scholar
  5. [33]
    E. Balas, W.R. Pulleyblank, The perfectly matchable subgraph polytope of an arbitrary graph. Combinatorica 9, 321–337 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  6. [37]
    I. Barany, T.J. Van Roy, L.A. Wolsey, Uncapacitated lot-sizing: the convex hull of solutions. Math. Program. 22, 32–43 (1984)zbMATHGoogle Scholar
  7. [50]
    C. Berge, Two theorems in graph theory. Proc. Natl. Acad. Sci. USA 43, 842–844 (1957)zbMATHMathSciNetCrossRefGoogle Scholar
  8. [69]
    R.D. Carr, G. Konjevod, G. Little, V. Natarajan, O. Parekh, Compacting cuts: new linear formulation for minimum cut. ACM Trans. Algorithms 5, 27:1–27:6 (2009)Google Scholar
  9. [71]
    M. Chudnovsky, G. Cornuéjols, X. Liu, P. Seymour, K. Vusković, Recognizing Berge graphs. Combinatorica 25, 143–186 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  10. [72]
    M. Chudnovsky, N. Robertson, P. Seymour, R. Thomas, The strong perfect graph theorem. Ann. Math. 164, 51–229 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  11. [79]
    M. Conforti, G. Cornuéjols, G. Zambelli, Extended formulations in combinatorial optimization. 4OR 8, 1–48 (2010)Google Scholar
  12. [81]
    M. Conforti, M. Di Summa, F. Eisenbrand, L.A. Wolsey, Network formulations of mixed-integer programs. Math. Oper. Res. 34, 194–209 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  13. [82]
    M. Conforti, L.A. Wolsey, Compact formulations as unions of polyhedra. Math. Program. 114, 277–289 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  14. [87]
    W.J. Cook, W.H. Cunningham, W.R. Pulleyblank, A. Schrijver, Combinatorial Optimization (Wiley, New York, 1998)zbMATHGoogle Scholar
  15. [89]
    W.J. Cook, J. Fonlupt, A. Schrijver, An integer analogue of Carathéodory’s theorem. J. Combin. Theory B 40, 63–70 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  16. [92]
    G. Cornuéjols, Combinatorial Optimization: Packing and Covering. SIAM Monograph, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 74 (2001)Google Scholar
  17. [103]
    G. Dantzig. R. Fulkerson, S. Johnson, Solution of a large-scale traveling-salesman problem. Oper. Res. 2, 393–410 (1954)Google Scholar
  18. [111]
    R. de Wolf, Nondeterministic quantum query and communication complexities. SIAM J. Comput. 32, 681–699 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  19. [120]
    E.A. Dinic, Algorithm for solution of a problem of maximum flow in networks with power estimation. Soviet Math. Dokl. 11, 1277–1280 (1970)Google Scholar
  20. [123]
    J. Edmonds, Paths, trees, and flowers. Can. J. Math. 17, 449–467 (1965)zbMATHMathSciNetCrossRefGoogle Scholar
  21. [124]
    J. Edmonds, Maximum matching and a polyhedron with 0,1-vertices. J. Res. Natl. Bur. Stand. B 69, 125–130 (1965)zbMATHMathSciNetCrossRefGoogle Scholar
  22. [126]
    J. Edmonds, Submodular functions, matroids, and certain polyhedra, in Combinatorial Structures and Their Applications, ed. by R. Guy, H. Hanani, N. Sauer, J. Schönheim. (Gordon and Breach, New York, 1970), pp. 69–87Google Scholar
  23. [128]
    J. Edmonds, R. Giles, A min-max relation for submodular functions on graphs. Ann. Discrete Math. 1, 185–204 (1977)MathSciNetCrossRefGoogle Scholar
  24. [129]
    J. Edmonds, R.M. Karp, Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM 19, 248–264 (1972)zbMATHCrossRefGoogle Scholar
  25. [136]
    S. Fiorini, S. Massar, S. Pokutta, H.R. Tiwary, R. de Wolf, Linear vs. semidefinite extended formulations: exponential separation and strong lower bounds, in STOC 2012 (2012)Google Scholar
  26. [137]
    S. Fiorini, V. Kaibel, K. Pashkovich, D.O. Theis Combinatorial bounds on the nonnegative rank and extended formulations. Discrete Math. 313, 67–83 (2013)Google Scholar
  27. [147]
    L.R. Ford Jr., D.R. Fulkerson, Flows in Networks (Princeton University Press, Princeton, 1962)zbMATHGoogle Scholar
  28. [148]
    A. Frank, Connections in combinatorial optimization, in Oxford Lecture Series in Mathematics and Its Applications, vol. 38 (Oxford University Press, Oxford, 2011)Google Scholar
  29. [154]
    D.R. Fulkerson, Blocking and anti-blocking pairs of polyhedra. Math. Program. 1, 168–194 (1971)zbMATHMathSciNetCrossRefGoogle Scholar
  30. [158]
    M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (W.H. Freeman and Co., New York, 1979)zbMATHGoogle Scholar
  31. [164]
    A.M.H. Gerards, A short proof of Tutte’s characterization of totally unimodular matrices. Linear Algebra Appl. 114/115, 207–212 (1989)Google Scholar
  32. [165]
    A. Ghouila-Houri, Caractérisation des matrices totalement unimodulaires. Comptes Rendus Hebdomadaires des Scéances de l’Académie des Sciences (Paris) 254, 1192–1194 (1962)zbMATHMathSciNetGoogle Scholar
  33. [166]
    F.R. Giles, W.R. Pulleyblank, Total dual integrality and integer polyhedra. Linear Algebra Appl. 25, 191–196 (1979)zbMATHMathSciNetCrossRefGoogle Scholar
  34. [170]
    M.X. Goemans, Smallest compact formulation for the permutahedron. Math. Program. Ser. A DOI 10.1007/s101007-014-0757-1 (2014)Google Scholar
  35. [182]
    J. Gouveia, P. Parrilo, R. Thomas, Lifts of convex sets and cone factorizations. Math. Oper. Res. 38, 248–264 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  36. [195]
    O. Günlük, Y. Pochet, Mixing mixed-integer inequalities. Math. Program. 90, 429–458 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  37. [199]
    I. Heller, C.B. Tompkins, An extension of a theorem of Dantzig’s, in Linear Inequalities and Related Systems, ed. by H.W. Kuhn, A.W. Tucker (Princeton University Press, Princeton, 1956), pp. 247–254Google Scholar
  38. [203]
    A.J. Hoffman, A generalization of max-flow min-cut. Math. Program. 6, 352–259 (1974)zbMATHCrossRefGoogle Scholar
  39. [209]
    S. Iwata, L. Fleischer, S. Fujishige, A combinatorial, strongly polynomial-time algorithm for minimizing submodular functions. J. ACM 48, 761–777 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  40. [211]
    R.G. Jeroslow, Representability in mixed integer programming, I: characterization results. Discrete Appl. Math. 17, 223–243 (1987)zbMATHMathSciNetGoogle Scholar
  41. [212]
    R.G Jeroslow, On defining sets of vertices of the hypercube by linear inequalities. Discrete Math. 11, 119–124 (1975)Google Scholar
  42. [213]
    R.G Jeroslow, J.K. Lowe, Modelling with integer variables. Math. Program. Stud. 22, 167–184 (1984)Google Scholar
  43. [219]
    V. Kaibel, Extended formulations in combinatorial optimization. Optima 85, 2–7 (2011)Google Scholar
  44. [220]
    V. Kaibel, K. Pashkovich, Constructing extended formulations from reflection relations, in Proceedings of IPCO XV O. Günlük, ed. by G. Woeginger. Lecture Notes in Computer Science, vol. 6655 (Springer, Berlin, 2011), pp. 287–300Google Scholar
  45. [221]
    V. Kaibel, K. Pashkovich, D.O. Theis, Symmetry matters for sizes of extended formulations. SIAM J. Discrete Math. 26(3), 1361–1382 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  46. [223]
    V. Kaibel, S. Weltge, A short proof that the extension complexity of the correlation polytope grows exponentially. arXiv:1307.3543 (2013)Google Scholar
  47. [230]
    D.R. Karger, Global min-cuts in RNC, and other ramifications of a simple min-cut algorithm, in Proceedings of SODA (1993), pp. 21–30Google Scholar
  48. [243]
    B. Korte, J. Vygen, Combinatorial Optimization: Theory and Algorithms (Springer, Berlin/Hidelberg, 2000)CrossRefGoogle Scholar
  49. [244]
    J.B. Kruskal Jr., On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Am. Math. Soc. 7, 48–50 (1956)zbMATHMathSciNetCrossRefGoogle Scholar
  50. [245]
    H.W. Kuhn, The Hungarian method for the assignment problem. Naval Res. Logistics Q. 2, 83–97 (1955)CrossRefGoogle Scholar
  51. [252]
    E. L. Lawler, Combinatorial Optimization: Networks and Matroids (Holt, Rinehart and Winston, New York, 1976)zbMATHGoogle Scholar
  52. [259]
    L. Lovász, Normal hypergraphs and the perfect graph conjecture. Discrete Math. 2, 253–267 (1972)zbMATHMathSciNetCrossRefGoogle Scholar
  53. [262]
    L. Lovász, M.D. Plummer, Matching Theory (Akadémiai Kiadó, Budapest, 1986) [Also: North Holland Mathematics Studies, vol. 121 (North Holland, Amsterdam)]zbMATHGoogle Scholar
  54. [269]
    R.K. Martin, Generating alternative mixed integer programming models using variable definition. Oper. Res. 35, 820–831 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  55. [270]
    R.K. Martin, Using separation algorithms to generate mixed integer model reformulations. Oper. Res. Lett. 10(3), 119–128 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  56. [271]
    R.K. Martin, R.L. Rardin, B.A. Campbell, Polyhedral characterization of discrete dynamic programming. Oper. Res. 38, 127–138 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  57. [276]
    R.R. Meyer, On the existence of optimal solutions to integer and mixed integer programming problems. Math. Program. 7, 223–235 (1974)zbMATHCrossRefGoogle Scholar
  58. [278]
    C.E. Miller, A.W. Tucker, R.A. Zemlin, Integer programming formulation of traveling salesman problems. J. ACM 7, 326–329 (1960)zbMATHMathSciNetCrossRefGoogle Scholar
  59. [282]
    H. Nagamochi, T. Ibaraki, Computing edge-connectivity in multiple and capacitated graphs. SIAM J. Discrete Math. 5, 54–66 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  60. [296]
    J. Oxley, Matroid Theory (Oxford University Press, New York, 2011)zbMATHCrossRefGoogle Scholar
  61. [304]
    J. Pap, Recognizing conic TDI systems is hard. Math. Program. 128, 43–48 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  62. [307]
    J. Petersen, Die Theorie der regulären graphs. Acta Matematica 15, 193–220 (1891)zbMATHCrossRefGoogle Scholar
  63. [308]
    Y. Pochet, L.A. Wolsey, Polyhedra for lot-sizing with Wagner–Whitin costs. Math. Program. 67, 297–324 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  64. [311]
    C.H. Papadimitriou, M. Yannakakis, On recognizing integer polyhedra. Combinatorica 10, 107–109 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  65. [313]
    A. Razborov, On the distributional complexity of disjointness. Theor. Comput. Sci. 106(2), 385–390 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  66. [317]
    T. Rothvoß, Some 0/1 polytopes need exponential size extended formulations. Math. Program. A 142, 255–268 (2012)CrossRefGoogle Scholar
  67. [318]
    T. Rothvoß, The matching polytope has exponential extension complexity, in Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC 2014), (2014), pp. 263–272Google Scholar
  68. [324]
    A. Schrijver, On total dual integrality. Linear Algebra Appl. 38, 27–32 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  69. [325]
    A. Schrijver, Theory of Linear and Integer Programming (Wiley, New York, 1986)zbMATHGoogle Scholar
  70. [326]
    A. Schrijver, A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. Combin. Theory Ser. B 80, 346–355 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  71. [327]
    A. Schrijver, Combinatorial Optimization: Polyhedra and Efficiency (Springer, Berlin, 2003)Google Scholar
  72. [329]
    P.D. Seymour, Decomposition of regular matroids. J. Combin. Theory B 28, 305–359 (1980)zbMATHMathSciNetCrossRefGoogle Scholar
  73. [333]
    M. Stoer, F. Wagner, A simple min-cut algorithm. J. ACM 44, 585–591 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  74. [338]
    K. Truemper, Matroid Decomposition (Academic, Boston, 1992)zbMATHGoogle Scholar
  75. [339]
    W.T. Tutte, A homotopy theorem for matroids I, II. Trans. Am. Math. Soc. 88, 905–917 (1958)MathSciNetGoogle Scholar
  76. [341]
    M. Van Vyve, The continuous mixing polyhedron. Math. Oper. Res. 30, 441–452 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  77. [342]
    F. Vanderbeck, L.A. Wolsey, Reformulation and decomposition of integer programs, in 50 Years of Integer Programming 1958–2008, ed. by M. Jünger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, L. Wolsey (Springer, New York, 2010), pp. 431–502Google Scholar
  78. [344]
    S. Vavasis, On the complexity of nonnegative matrix factorization. SIAM J. Optim. 20, 1364–1377 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  79. [347]
    J.P. Vielma, Mixed integer linear programming formulation techniques to appear in SIAM Review (2014)Google Scholar
  80. [355]
    M. Yannakakis, Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci. 43, 441–466 (1991)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Michele Conforti
    • 1
  • Gérard Cornuéjols
    • 2
  • Giacomo Zambelli
    • 3
  1. 1.Department of MathematicsUniversity of PadovaPadovaItaly
  2. 2.Tepper School of BusinessCarnegie Mellon UniversityPittsburghUSA
  3. 3.Department of ManagementLondon School of Economics and Political ScienceLondonUK

Personalised recommendations