All linear and integer programs are slim 3-way transportation programs

SIAM Journal on Optimization

Volumn 17, Issue 3, 2006, Pages 806-821

  De Loera, Jesús A ^a   Onn, Shmuel ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

Author keywords

Combinatorial optimization; Contingency tables; Convex polytopes; Data security; Integer programming; Linear programming; Multicommodity flows; Multiway table; Privacy; Statistical table; Strongly polynomial time; Transportation problems

Indexed keywords

COMBINATORIAL OPTIMIZATION; DATA PRIVACY; LINEAR PROGRAMMING; POLYNOMIALS; PROBLEM SOLVING; STATISTICAL METHODS;

CONTINGENCY TABLES; CONVEX POLYTOPES; MULTICOMMODITY FLOWS; MULTIWAY TABLES; STATISTICAL TABLES;

INTEGER PROGRAMMING;

EID: 34547960144     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/040610623     Document Type: Article

References (30)

1. M. BAIÖU AND M. L. BALINSKI, The stable allocation (or ordinal transportation) problem, Math. Oper. Res., 27 (2002), pp. 485-503.
2. M. L. BALINSKI AND F. J. RISPOLI, Signature classes of transportation polytopes, Math. Program Ser. A, 60 (1993), pp. 127-144.
3. L. H. COX, Bounds on entries in 3-dimensional contingency tables, in Inference Control in Statistical Databases: From Theory to Practice, J. Domingo-Ferrer, ed., Lecture Notes in Comput. Sci. 2316, Springer, New York, 2002, pp. 21-33.
4. L. H. COX, On properties of multi-dimensional statistical tables, J. Stat. Plann. Inference, 117 (2003), pp. 251-273.
5. M. CRYAN, M. DYER, H. MÜLLER, AND L. STOUGIE, Random walks on the vertices of transportation polytopes with constant number of sources, in Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms (Baltimore, MD), ACM, New York, 2003, pp. 330-339.
6. J. DE LOERA AND S. ONN, The complexity of three-way statistical tables, SIAM J. Comput., 33 (2004), pp. 819-836.
7. J. DE LOERA AND S. ONN, All rational polytopes are transportation polytopes and all polytopal integer sets are contingency tables, in Integer Programming and Combinatorial Optimization, Proceedings of the 10th International IPCO Conference, Lecture Notes in Comput. Sei. 3064, Springer, New York, 2004, pp. 338-351.
8. J. DE LOERA AND S. ONN, Markov bases of three-way tables are arbitrarily complicated, J. Symbolic Comput., 41 (2006), pp. 173-181.
9. P. DIACONIS AND A. GANGOLLI, Rectangular arrays with fixed margins in Discrete Probability and Algorithms (Minneapolis, MN, 1993), D. Aldous, P. Diaconis, J. Spencer, and J. M. Steele, eds., IMA Vol. Math. Appl. 72, Springer, New York, 1995, pp. 15-41.
10. G. T. DUNCAN, S. E. FIENBERG, R. KRISHNAN, R. PADMAN, AND S. F. ROEHRIG, Disclosure limitation methods and information loss for tabular data, in Confidentiality, Disclosure and Data Access: Theory and Practical Applications for Statistical Agencies, P. Doyle, J, I. Land, J. M. Theeuwes, and L. V. Zayatz, eds., North-Holland, Amsterdam, 2001, pp. 135-166.
11. L. K. FLEISCHER, Approximating fractional multicommodity flow independent of the number of commodities, SIAM J. Discrete Math., 13 (2000), pp. 505-520.
12. N. GARG AND J. KÖNEMANN, Faster and simpler algorithms for multicommodity flow and other fractional packing problems, in Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science (New York), IEEE, New York, 1998, pp. 300-309.
13. D. GUSFIELD, A graph theoretic approach to statistical data security, SIAM J. Comput., 17 (1988), pp. 552-571.
14. 1-ε, Acta Math., 182 (1999), pp. 105-142.
15. S. HOŞTEN AND S. SULLIVANT, Gröbner bases and polyhedral geometry of reducible and cyclic models, J. Combin. Theory Ser. A, 100 (2002), pp. 277-301.
16. T. C. HU, Multi-commodity network flows, Oper. Res., 11 (1963), pp. 344-360.
17. F. K. HWANG, S. ONN, AND U. G. ROTHBLUM, A polynomial time algorithm for shaped partition problems, SIAM J. Optim., 10 (1999), pp. 70-81.
18. R. IRVING AND M. R. JERRUM, Three-dimensional statistical data security problems, SIAM J. Comput., 23 (1994), pp. 170-184.
19. A. ITAI, Two-commodity flow, J. Assoc. Comput. Mach., 25 (1978), pp. 596-611.
20. V. KLEE AND C. WITZGALL, Facets and vertices of transportation polytopes, in Mathematics of the Decision Sciences, Part I (Stanford, CA, 1967), AMS, Providence, RI, 1968, pp. 257-282.
21. A. LEVIN, Personal communication, 2004.
22. C. R. MEHTA AND N. R. PATEL, A network algorithm for performing Fisher's exact test in r x c contingency tables, J. Amer. Statist. Assoc., 78 (1983), pp. 427-434.
23. S. ONN AND U. G. ROTHBLUM, Convex combinatorial optimization, Discrete Comput. Geom., 32 (2004), pp. 549-566.
24. S. ONN AND L. J. SCHULMAN, The vector partition problem for convex objective functions, Math. Oper. Res., 26 (2001), pp. 583-590.
25. A. SCHRIJVER, Theory of Linear and Integer Programming, John Wiley &c Sons, Ltd., Chichester, 1986.
26. E. TARDOS, A strongly polynomial algorithm to solve combinatorial linear programs, Oper. Res., 34 (1986), pp. 250-256.
27. UNIVERSAL GENERATOR, http://www.math.ucdavis.edu/ ~deloera, http://ie.technion.ac.il/~onn.
28. U.S. BUREAU OF THE CENSUS, American FactFinder, interactive information accessible at http://factfinder.census.gov/servlet/ BasicFactsServlet.
29. M. VLACH, Conditions for the existence of solutions of the three-dimensional planar transportation problem, Discrete Appl. Math., 13 (1986), pp. 61-78.
30. V. A. YEMELICHEV, M. M. KOVALEV, AND M. K. KRAVTSOV, Polytopes, Graphs and Optimisation, Cambridge University Press, Cambridge, UK, 1984.