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Discrete subadditive functions as Gomory functions

Published online by Cambridge University Press:  24 October 2008

Frank Rhodes  and
H. Paul Williams
Frank Rhodes
Affiliation:
Department of Mathematics, University of Southampton, Southampton SO9 5NH
H. Paul Williams
Affiliation:
Department of Mathematics, University of Southampton, Southampton SO9 5NH
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Extract

Our aim, in this paper, is to study a class of functions which occurs in pure integer programming, and to investigate conditions under which discrete subadditive functions belong to that class. The inspiration for the paper was the problem of classifying discrete metrics used in pattern recognition, while the methods of proof of the main theorem are those of pure integer programming.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 117 , Issue 3 , May 1995 , pp. 559 - 574
Copyright
Copyright © Cambridge Philosophical Society 1995

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References

REFERENCES

[1]Borgefors, G.. Distance transformations in arbitrary dimensions. Computer Vision, Graphics, and Image Processing 27 (1984) 321–345.CrossRefGoogle Scholar
[2]Blair, C. E. and Jeroslow, R. G.. The value function of an integer program. Mathematical Programming 23 (1982), 237–273.CrossRefGoogle Scholar
[3]Chaudhuri, D., Murthy, C. A. and Chaudhuri, B. B.. A modified metric to compute distance. Pattern Recognition 25 (1992), 667–677.CrossRefGoogle Scholar
[4]Chvátal, V.. Edmonds polynomials and a hierarchy of combinatorial problems. Discrete Mathematics 4 (1973), 305–337.CrossRefGoogle Scholar
[5]Garfinkel, R. S. and Nemhauser, G. L.. Integer programming (John Wiley & Sons, 1972).Google Scholar
[6]Gomory, R. E.. An algorithm for integer solutions to linear programs. In: Recent Advances in Mathematical Programming, Graves, R. and Wolf, P., Eds. (McGraw Hill, 1963), 269–302.Google Scholar
[7]Gomory, R. E.. On the relation between integer and non-integer solutions to linear programs. Proc. Nat. Acad. Sci. 53, (1965), 260–265.CrossRefGoogle Scholar
[8]Gomory, R. E.. Some polyhedra related to combinatorial problems. Linear Algebra and its Applications 2 (1969), 451–558.CrossRefGoogle Scholar
[9]Hilbert, D.. Über die Theorie der algebraischen Formen, Math. Ann. 36 (1890), 475–534.CrossRefGoogle Scholar
[10]Nemhauser, G. L. and Wolsey, L. A.. Integer and Combinatorial Optimization (Wiley-Interscience, 1988).CrossRefGoogle Scholar
[11]Rhodes, F.. Discrete metrics as Gomory functions. In: Vision Geometry II, Melter, R. A. and Wu, A. Y., Eds. (Proc. SPIE 2060) 1993, 89–95.CrossRefGoogle Scholar
[12]Rhodes, F.. On the metrics of Chaudhuri, Murthy and Chaudhuri, Pattern Recognition, to appear.Google Scholar
[13]Rhodes, F.. Metric subgraphs of the chamfer metrics and the Melter-Tomescu path generated metrics, Discrete Mathematics, to appear.Google Scholar
[14]Schrijver, A.. Theory of linear and integer programming (Wiley, 1986).Google Scholar
[15]Williams, H. P.. An alternative form of the value function of an integer programme, Working Paper OR16, Faculty of Mathematical Studies, University of Southampton, Southampton SO9 5NH, England, 1988.Google Scholar
[16]Williams, H. P.. Constructing the value function for an integer linear programme over a cone, Computational Optimization and Applications, to appear.Google Scholar
[17]Wolsey, L. A.. Integer programming duality: Price functions and sensitivity analysis. Mathematical Programming 20 (1981), 173–195.CrossRefGoogle Scholar
[18]Wolsey, L. A.. The b-hull of an integer programme. Discrete Applied Mathematics 3 (1981), 193–201.CrossRefGoogle Scholar