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Directed hypergraphs as a modelling paradigm

  • Giorgio GalloEmail author
  • Maria Grazia Scutellà
Article

Abstract

We address a generalization of graphs, the directed hypergraphs, and show that they are a powerful tool in modelling and solving several relevant problems in many application areas. Such application areas include Linear Production Problems, Flexible Manufacturing Systems, Propositional Logic, Relational Databases, and Public Transportation Systems.

Keywords

Flexible Manufacture System Simplex Algorithm Directed Cycle Context Free Language Variable Symbol 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Gli ipergrafi orientati come strumento per la modellazione e la soluzione di problemi decisionali

Riassunto

Gli ipergrafi orientati costitniscono uno strumento di notevole efficacia nella modellazione e soluzione di diverse classi di problemi decisionali. Essi costituiscono un linguaggio unificante per problemi che nascono in ambienti molto diversi, quali “Sistemi di Produzione”, “Sistemi di Trasporto Pubblico”, “Basi di Dati Relazionali” …

Particolarmente importanti sono i concetti di ipercammino e di iperflusso che generlizzano in modo naturale quelli di cammino e di flusso su grafi orientati. fpercammini e iperflussi ottimi possono essere trovati per mezzo di algoritmi specializzati, che generalizzano gli algoritmi di etichettatura per la ricerca di cammini minimi e quelli di tipo “simplesso su reti” per i flussi di costo minimo.

Ci si propone di presentare alcuni dei contesti più rilevanti, dal punto di vista delle applicazioni pratiche, in cui possono essere utilizzati modelli e algoritmi basati su ipergrafi orientati. Particolare attenzione sarà rivolta alle applicazioni di tipo economico aziendale.

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References

  1. [1]
    Bala M., Martin K.,A mathematical programning approach to data base normalization, Technical report, Graduate School of Business, University of Chicago, 1993.Google Scholar
  2. [2]
    Basu A., Blanning R. W.,A tool for modeling decision support systems, Management Science, 40 (12), 1994, 1579–1600.CrossRefGoogle Scholar
  3. [3]
    Berge C.,Graphs and Hypergraphs, North-Holland, Amsterdam, 1973.Google Scholar
  4. [4]
    Berge C.,Minimax theorems for normal hypergraphs and balanced hyper-graphs—a survey, Annals of Discrete Mathematics, 21, 1984, 3–19.Google Scholar
  5. [5]
    Berce C.,Hypergraphs: Combinatorics of Finite Sels, North-Holland, Amsterdam, 1989.Google Scholar
  6. [6]
    Boley H.,Directed recursive label node hyeergraphs: a new representation language, Artificial Intelligence, 9, 1977, 49–85.CrossRefGoogle Scholar
  7. [7]
    Cook S.,The complexity of theorem-proving procedures, in Proc. 3-th acm symp. on Theory of Computing, 1971, 151–158.Google Scholar
  8. [8]
    Dowling W., Gallier J.,Linear-time algorithms for testing the satisfiability of propositional Horn formulae, J. Logic Programming, 3, 1984, 267–284.CrossRefGoogle Scholar
  9. [9]
    Dyckhoff H.,A typology of cutting and packing problems, European J. Oper. Res., 44, 1990, 145–159.CrossRefGoogle Scholar
  10. [10]
    Furtado A. L.,Formal aspects of the relational model, Information Systems, 3, 1978, 131–140.CrossRefGoogle Scholar
  11. [11]
    Gale D.,The Theory of Linear Economic Models, McGraw-Hill, New York, NY, 1960.Google Scholar
  12. [12]
    Gallo G., Gentile C., Pretolani D., Rago G.,Max Horn Sat and the minimum cut problem on directed hypergaphs, Math. Programming, 80, 1998, 213–237.Google Scholar
  13. [13]
    Gallo G., Licheri N., Scutellà M. G.,The Hypergraph Simplex approach: some experimental results, Ricerca Operativa, 78 (anno XXVI), 1996, 21–54.Google Scholar
  14. [14]
    Gallo G., Longo G., Nguyen S., Pallottino S. Directed hypergraphs and applications, Discrete Appl. Math., 40, 1993, 177–201.CrossRefGoogle Scholar
  15. [15]
    Gallo G., Pallottino S.,Hypergraph models and algorithms for the Assembly Problem, Technical Report TR-6/92, Dipartimento di Informatica, Università di Pisa, Pisa, Italy, 1992.Google Scholar
  16. [16]
    Gallo G., Scutellà, M. G.,Minimum makespan assembly plans, Technical Report TR-98-10, Dipartimento di Informatica, Università di Pisa, Pisa, Italy, 1998.Google Scholar
  17. [17]
    Gallo G., Urbani G.,Algorithms for testing the satisfiability of propositional formulae, J. Logic Programming, 7, 1989, 45–61.CrossRefGoogle Scholar
  18. [18]
    Gambale M., Nonato M., Scutellà M. G.,The cutting stock problem: a new model based on hypergraph flows, Ricerca Operativa, 74 (anno XXV), 1995, 73–97.Google Scholar
  19. [19]
    Garey M. R., Johnson D. S.,Computers and Intractability: A Guide to the Theory of NP-completcness, W. H. Freeman, San Francisco, CA, 1979.Google Scholar
  20. [20]
    Gnesi S., Montanari U., Martelli A.,Dynamic programming as graph searching: an algebraic approach, J. Assoc. Comput. Mach, 28, 1981, 737–751.Google Scholar
  21. [21]
    Henderson J. M., Quandt R. E.,Microeconomic Theory, a Mathematical approach, McGraw-Hill, New York, NY, 1971.Google Scholar
  22. [22]
    Hopcroft J. E., Ullman J. D.,Formal languages and their relation to automata. Addison-Wesley, 1969.Google Scholar
  23. [23]
    Itai A., Makowsky J.,On the complexity of Herbrand’s theorem, Technical Report T. R. 213, Dept. Comp. Sci., Israel Inst. of Technology, Israel, 1982.Google Scholar
  24. [24]
    Italiano G. F., Nanni U.,On line maintenance of minimal directed hypergraphs, in Prec. 3 Convegno Italiano di Informatica Teorica, World Science Press, 1989, 335–349.Google Scholar
  25. [25]
    Jaumard B., Simeone B.,On the complexity of the maximum satisfiability problem for Horn formulas, Inform. Process. Lett., 26, 1987, 1–4.CrossRefGoogle Scholar
  26. [26]
    Jeroslow R. G., Martin R. K., Rardin R. R., Wang G.,Gainfree Leontief substitution flow problems, Math. Programming, 57, 1992, 375–414.CrossRefGoogle Scholar
  27. [27]
    Knuth D. E.,A generatization of Dijkstra’s algorithm, Inform. Process. Lett., 6 (1), 1977, 1–5.CrossRefGoogle Scholar
  28. [28]
    Levi G., Sirovich F.,Generulized And/Or Graphs, Artificial Intelligence, 7, 1976, 243–259.CrossRefGoogle Scholar
  29. [29]
    Lucchesi C. L., Osborn S. L.,Candidate keys for relations, J. Comput. System Sci., 17, 1978, 270–279.CrossRefGoogle Scholar
  30. [30]
    Maier D.,Minimum covers in the relational data base model, J. Assoc. Comput. Mach., 27, 1980, 664–674.Google Scholar
  31. [31]
    Marcotte P., Nguyen S.,Hyperpath formulations of traffic assignment problems, in P. Marcotte and S. Nguyen, editors,Equilibrium and Advanced Transportation Modelling, Kluwer Academic, MA, 1998, 175–200.Google Scholar
  32. [32]
    Martelli A., Montanari U.,Additive And/Or graphs, Proceeding IJCAI, 3, 1977, 1–11.Google Scholar
  33. [33]
    Homen De Mello L. S., Sanderson A. C.,And/Or graph representation of assenbly plans, IEEE Transactions on Robotics and Automation, 6, 1990, 188–199.CrossRefGoogle Scholar
  34. [34]
    Nguyen S., Pallottino S. Equilibrium traffic assignment for large scale transit network, European J. Oper. Res., 37, 1988, 176–186.CrossRefGoogle Scholar
  35. [35]
    Nguyen S., Pallottino S. Hyperpaths and shortest hyperpaths, in B. Simeone, editor,Combinatorial Optimization, volume 1403 ofLectures Notes in Mathematics, Springer-Verlag, Berlin, 1989, 258–271.Google Scholar
  36. [36]
    Nilsson N. J.,Problem Solving Methods in Artificial Intelligence, McGraw-Hill, New York, NY, 1971.Google Scholar
  37. [37]
    Nilsson N. J.,Principles of Artificial Intelligence, Morgan Kaufmann, Los Altos, CA, 1980.Google Scholar
  38. [38]
    Cambini R., Gallo G., Scutella M. G.,Flows on hypergraphs, Math. Programming, 78, 1997, 195–217.Google Scholar
  39. [39]
    Scheithauer G., Terno H.,About the gap between the optimal values of the integer and continnous relaxation one-dimensional cutting stock prblem, in Oper. Res. Proc., Springer Verlag, Berlin 1992.Google Scholar
  40. [40]
    Scheithauer G., Terno H.,The modified integer round-up property for the one-dimensional cutting stock problem, European J. Oper. Res., 84, 1995, 562–571.CrossRefGoogle Scholar
  41. [41]
    Scheithauer G., Terno H.,Theoretical investigations on the modified integer round-up property for the one-dimensional cutting stock problem, Oper. Res. Lett., 20, 1997, 93–100.CrossRefGoogle Scholar
  42. [42]
    Takayama A.,Mathematical Economics, Cambridge University Press, 1985.Google Scholar
  43. [43]
    Ullman J. D.,Principles of Database Systems, Computer Science Press, Reckville, MD, 1982.Google Scholar

Copyright information

© Associazione per la Matematica Applicata alle Scienze Economiche e Sociali (AMASES) 1998

Authors and Affiliations

  1. 1.Dipartimento di InformaticaUniversità di PisaPisa

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