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Heidelberg Colloquium on Glassy Dynamics pp 325–353Cite as

Calculating exact ground states of spin glasses: A polyhedral approach

  • II. Optimization
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Part of the book series: Lecture Notes in Physics ((LNP,volume 275))

Abstract

In this paper we describe recent developments in the theory and algorithm design of combinatorial optimization that are related to questions concerning ground states of spin glasses. In particular, we outline an approach, based on polyhedral combinatorics, that has led to the implementation of a cutting plane method for calculating exact ground states of spin glasses in the Ising model. With this method exact ground states for planar grids of size up to 40 x 40 with periodic boundary conditions and exterior magnetic field can be determined in reasonable running times.

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References

  • J. C. Angles d'Auriac & R. Maynard (1984), “On the random antiphase state in the ± J spin glass model in two dimensions”, Solid State Communications 49 (1984) 785–790.

    Article  Google Scholar 

  • F. Barahona (1981), “Balancing signed toroidal graphs in polynomial time”, Departamento de Matemáticas, Universidad de Chile, 1981.

    Google Scholar 

  • F. Barahona (1982), “On the computational complexity of Ising spin glass models”, J. Phys. A: Math. Gen. 15 (1982) 3241–3253.

    Article  MathSciNet  ADS  Google Scholar 

  • F. Barahona (1983), “The max cut problem in graphs not contractible to K5”, Operations Research Letters 2 (1983) 107–111.

    Article  MATH  MathSciNet  Google Scholar 

  • F. Barahona (1985), “Finding ground states in random-field Ising ferromagnets”, J. Phys. A: Math. Gen. 18 (1985) L673–L675.

    Article  MathSciNet  ADS  Google Scholar 

  • F. Barahona, M. Grötschel & A. R. Mahjoub (1985), “Facets of the bipartite subgraph polytope”, Mathematics of Operations Research 10 (1985) 340–358.

    Article  MATH  MathSciNet  Google Scholar 

  • F. Barahona, M. Jünger, M. Grötschel & G. Reinelt (1986), “An application of combinatorial optimization to statistical physics and circuit layout design”, Preprint no. 95, Institut für Mathematik, Universität Augsburg, 1986.

    Google Scholar 

  • F. Barahona & E. Maccioni (1982), “On the exact ground states of three-dimensional Ising spin glasses”, J. Phys. A. Math. Gen. 15 (1982) L611–L615.

    Article  MathSciNet  ADS  Google Scholar 

  • F. Barahona & A. R. Mahjoub (1983), “On the cut polytope”, Report No. 83 271OR, Institut für Ökonometrie and Operations Research, Universität Bonn, Bonn, W. Germany, 1983.

    Google Scholar 

  • F. Barahona, R. Maynard, R. Rammal & J. P. Uhry (1982), “Morphology of ground states of a two dimensional frustration model”, J. Phys. A. Math. Gen. 15 (1982) L673–L699.

    Article  MathSciNet  ADS  Google Scholar 

  • H. Crowder, E. L. Johnson & M. Padberg (1983), “Solving large-scale zero-one linear programming problems”, Operations Research 31 (1983) 803–834.

    Article  MATH  Google Scholar 

  • H. Crowder & M. Padberg (1980), “Solving large-scale symmetric travelling salesman problems to optimality”, Management Science 26 (1980) 495–509.

    Article  MATH  MathSciNet  Google Scholar 

  • G. B. Dantzig, D. R. Fulkerson & S. M. Johnson (1954), “Solution of a large-scale traveling-salesman problem”, Operations Research 2 (1954) 393–410.

    Article  MathSciNet  Google Scholar 

  • A. Dress (1986), “Computing spin-glass Hamiltonians”, Preprint, Universität Bielefeld, 1986.

    Google Scholar 

  • J. Edmonds & E. L. Johnson (1973), “Matching, Euler tours, and the Chinese postman”, Mathematical Programming 5 (1973) 88–124.

    Article  MATH  MathSciNet  Google Scholar 

  • J. Fonlupt, A. R. Mahjoub & J. P. Uhry (1984), “Composition of graphs and the bipartite subgraph polytope”, R. R. no. 459, Laboratoire ARTEMIS (IMAG), Université de Grenoble, 1984.

    Google Scholar 

  • M. R. Garey & D. S. Johnson (1979), Computers and Intractability: A Guide to the Theory of NP-completeness, Freeman, San Francisco, 1979.

    MATH  Google Scholar 

  • A. M. H. Gerards (1985), “Testing the odd bicycle wheel inequalities for the bipartite subgraph polytope”, Mathematics of Operations Research 10 (1985) 359–360.

    Article  MATH  MathSciNet  Google Scholar 

  • M. Grötschel (1977), Polyedrische Charakterisierungen kombinatorischer Optimierungsprobleme, Verlag Anton Hain, Meisenheim am Glan, (1977).

    MATH  Google Scholar 

  • M. Grötschel & O. Holland (1985), “Solving matching problems with linear programming”, Mathematical Programming 33 (1985) 243–259.

    Article  MATH  MathSciNet  Google Scholar 

  • M. Grötschel, M. Jünger & G. Reinelt (1984), “A cutting plane algorithm for the linear ordering problem”, Operations Research 32 (1984) 1195–1220.

    Article  MATH  MathSciNet  Google Scholar 

  • M. Grötschel, L. Loväsz & A. Schrijver (1981), “The ellipsoid method and its consequences in combinatorial optimization”, Combinatorica 1 (1981) 169–197.

    Article  MATH  MathSciNet  Google Scholar 

  • M. Grötschel, L. Loväsz & A. Schrijver (1987), Geometric Algorithms and Combinatorial Optimization, Springer, Heidelberg, 1987, (to appear).

    Google Scholar 

  • M. Grötschel & G. L. Nemhauser (1984), “A polynomial algorithm for the max-cut problem on graphs without long odd cycles”, Mathematical Programming 29 (1984) 28–40.

    Article  MATH  MathSciNet  Google Scholar 

  • M. Grötschel & M. W. Padberg (1979a), “On the symmetric travelling salesman problem I: inequalities”, Mathematical Programming 16 (1979) 265–280.

    Article  MATH  MathSciNet  Google Scholar 

  • M. Grötschel & M. W. Padberg (1979b), “On the symmetric travelling salesman problem II: lifting theorems and facets”, Mathematical Programming 16 (1979) 281–302.

    Article  MATH  MathSciNet  Google Scholar 

  • M. Grötschel & W. R. Pulleyblank (1981), “Weakly bipartite graphs and the max-cut problem”, Operations Research Letters 1 (1981) 23–27.

    Article  MATH  MathSciNet  Google Scholar 

  • M. Grötschel & K. Truemper (1986), “Decomposition and optimization over cycles in binary matroids”, Preprint no. 108, Institut für Mathematik, Universität Augsburg, 1986.

    Google Scholar 

  • F. Hadlock (1975), “Finding a maximum cut of a planar graph in polynomial time”, SIAM Journal on Computing 4 (1975) 221–225.

    Article  MATH  MathSciNet  Google Scholar 

  • A. Hartwig, F. Daske & S. Kobe (1984), “A recursive branch-and-bound algorithm for the exact ground state of Ising spin-glass models”, Computer Physics Communications 32 (1984) 133–138.

    Article  ADS  Google Scholar 

  • R. M. Karp (1972), “Reducibility among combinatorial problems”, in: R. E. Miller & J. W. Thatcher (eds.), Complexity of Computer Computations, Plenum Press, New York, 1972, 85–103.

    Google Scholar 

  • E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan & D. Shmoys (1985), The Traveling Salesman Problem, Wiley, Chichester, 1985.

    MATH  Google Scholar 

  • I. Morgenstern (1983), “Numerical simulations of spin glasses”, in: J. L. van Hemmen & I. Morgenstern (eds.), Heidelberg Colloquium on Spin Glasses, Lecture Notes in Physics no. 192, Springer, Berlin, 1983, 305–327.

    Chapter  Google Scholar 

  • I. Morgenstern & K. Binder (1980), “Magnetic correlations in two-dimensional spinglasses”, Physical Review B 22 (1980) 288–302.

    Article  ADS  Google Scholar 

  • G. I. Orlova & Y. G. Dorfman (1972), ‘Finding the maximal cut in a graph”, Engrg. Cybernetics 10 (1972) 502–506.

    MATH  MathSciNet  Google Scholar 

  • J. C. Picard & H. D. Ratliff (1975), “Minimum cuts and related problems”, Networks 5 (1975) 357–370.

    Article  MATH  MathSciNet  Google Scholar 

  • G. Toulouse (1977), “Theory of the frustration effect in spin glasses: I”, Communications on Physics 2 (1977) 115–119.

    Google Scholar 

  • G. Toulouse (1983), “Frustration and disorder, new problems in statistical mechanics, spin glasses in a historical perspective”, in: J. L. van Hemmen & I. Morgenstern (eds.), Heidelberg Colloquium on Spin Glasses, Lecture Notes in Physics no. 192, Springer, Berlin, 1983, 2–17.

    Chapter  Google Scholar 

  • K. Wagner (1937), “Über eine Erweiterung des Satzes von Kuratowski”, Deutsche Mathematik 2 (1937) 280–285.

    MATH  Google Scholar 

  • M. Yannakakis (1978), “Node-and edge-deletion NP-complete problems”, Proc. 10th Ann. ACM Symposium on Theory of Computing, Association of Computing Machinery, New York, 1978, 253–264.

    Google Scholar 

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Authors and Affiliations

  1. Institut für Mathematik, Universität Augsburg, Memminger Str. 6, D-8900, Augsburg, FRG

    M. Grötschel, M. Jünger & G. Reinelt

Authors
  1. M. Grötschel
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  2. M. Jünger
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  3. G. Reinelt
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J. L. van Hemmen I. Morgenstern

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© 1987 Springer-Verlag

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Grötschel, M., Jünger, M., Reinelt, G. (1987). Calculating exact ground states of spin glasses: A polyhedral approach. In: van Hemmen, J.L., Morgenstern, I. (eds) Heidelberg Colloquium on Glassy Dynamics. Lecture Notes in Physics, vol 275. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0057526

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  • DOI: https://doi.org/10.1007/BFb0057526

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-17777-7

  • Online ISBN: 978-3-540-47819-5

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