Skip to main content
Log in

Some problems in discrete optimization

  • Published:
Mathematical Programming Submit manuscript

Abstract

The present paper concentrates on several problems of network flows and discrete optimization. Progress has been made on some of the problems while little is known about others. Some of the problems discussed are shortest paths, multi-commodity flows, traveling salesman problems, m-center problem, telepak problems and binary trees.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. C. Berge,The theory of graphs, Translated by A. Doig (Wiley, New York, 1962).

    Google Scholar 

  2. C. Berge and A. Ghouila-Houri,Programming, games and transportation networks. translated by M. Merrington and C. Ramanujacharyula (Wiley, New York, 1965).

    Google Scholar 

  3. R.G. Busacker and T.L. Saaty,Finite graphs and networks (McGraw-Hill, New York, 1965).

    Google Scholar 

  4. N. Deo,An extensive English language bibliography on graph theory and its applications, Technical Report 32-1413, Jet Propulsion Laboratory (October 1969).

  5. L.R. Ford, Jr. and D.R. Fulkerson,Flows in networks (Princeton University Press, 1962).

  6. H. Frank and I.T. Frisch,Communication, transmission and transportation networks (Addison-Wesley, 1971).

  7. D.R. Fulkerson, “Flow networks and combinatorial operations research”,American Mathematical Monthly 73 (2) (1966) 115–138.

    Google Scholar 

  8. F. Harary,Graph theory (Addison-Wesley, New York, 1969).

    Google Scholar 

  9. Graph theory and its applications, ed. B. Harris (Academic Press, New York, 1970).

  10. T.C. Hu,Integer programming and network flows (Addison-Wesley, New York, 1969).

    Google Scholar 

  11. M. Iri,Network flow, transportation and scheduling (Academic Press, New York, 1969).

    Google Scholar 

  12. D.E. Knuth,The art of computer programming, Vol. 1:Fundamental algorithms (Addison-Wesley, New York, 1968).

    Google Scholar 

  13. J.W. Moon,Topics on tournaments (Holt, Rinehart and Winston, New York, 1968).

    Google Scholar 

  14. O. Ore,Theory of graphs (American Math. Soc. Colloquium Publication, 1962).

  15. RAND “Research in combinatorics”, a bibliography of selected RAND Publications, RAND Report Dept. (Jan. 1970).

  16. J. Turner and W. Kautz, “A survey of progress in graph theory in the Soviet Union”,SIAM Review 12 (1970) (Supplement).

  17. J. Turner, “Key-word indexed bibliography of graph theory” In:Proof techniques in graph theory, ed. F. Harary (Academic Press, New York, 1969) pp. 189–330.

    Google Scholar 

  18. M. Bellmore and H.D. Ratliff, “Set covering and involutory bases”, The Johns Hopkins University, Report (1970).

  19. M. Bellmore and H.D. Ratliff, “Optimal defense of multi-commodity networks”, to appear inManagement Science (Applications).

  20. J. Edmonds and R.M. Karp, “A labeling method for maximal network flows which is bounded by a polynomial in the number of nodes”, to appear. (Reported in T.C. Hu,Integer programming and network flows, pp. 161–170.)

  21. D.J. Kleitman, “Methods for investigating connectivity of larger graphs”,IEEE Transactions on Circuit Theory (May, 1969).

  22. D.C. Plisch, “New results concerning separation theory of graphs”, Ph.D. Thesis, E.E. Dept., University of Wisconsin, 1970 (under Professor D.P. Brown).

  23. E.W. Dijkstra, “A note on two problems in connection with graphs”,Numerische Mathematik 1 (1959) 265–271.

    Google Scholar 

  24. S.E. Dreyfus, “An appraisal of some shortest path algorithms”,Operations Research 17, No. 3 (1965).

    Google Scholar 

  25. M. Florian and P. Robert, “A direct search method to locate negative cycles in a graph with application to minimum cost flows”, Pub. # 14, Dept. of Information, University of Montreal, Canada.

  26. R.W. Floyd, “Algorithm 97: Shortest path”,Communication of ACM 5 (6) (1962) 345.

    Google Scholar 

  27. T.C. Hu, “A decomposition algorithm for shortest paths in a network”.Operations Research 16 (1) (1968) 91–102.

    Google Scholar 

  28. T.C. Hu and W.T. Torres, “Shortcut in the decomposition algorithm for shortest paths in a network”,IBM Journal of Research and Development 13 (4) (1969) 387–390.

    Google Scholar 

  29. A.H. Land and S.W. Stairs, “The extension of the cascade algorithm to larger graphs”,Management Science 14 (L) (1967) pp. 29–33.

    Google Scholar 

  30. J.D. Murchland, “Bibliography of the shortest route problem”, LBS-TNT-62 (March, 1969); JDM-134 (July, 1970); Institut für Angewandte Reaktorphysik, Kernforschungszentrum Karlsruhe, Germany.

  31. S. Warshall, “A theorem on Boolean matrices”,Journal of ACM 9 (11–12) (1962).

  32. J.Y. Yen, “A decomposition algorithm for finding all shortest distances in a series of linearly overlapping networks”, to appear inOperations Research.

  33. L.R. Ford, Jr. and D.R. Fulkerson, “Suggested computation for maximal multicommodity network flows”,Management Sciences 5 (1) (1958) 97–101.

    Google Scholar 

  34. R.E. Gomory and T.C. Hu, “Synthesis of a communication network”,Journal of SIAM 12 (2) (1964) 348–369. (Also reported in T.C. Hu,Integer programming and network flows, pp. 197–213.)

    Google Scholar 

  35. R.C. Grinold, “A multi-commodity max flow algorith”,Operations Research 16 (6) (1968) pp. 1235–1238.

    Google Scholar 

  36. J.K. Hartman and L.S. Lasdon, “A generalized upper bounding algorithm for multicommodity network flow problems'., Tech. Memo No. 193, Case Western Reserve Univ. (June, 1970).

  37. T.C. Hu, “Multi-commodity network flows”,Operations Research 11 (3) (1963) 344–360.

    Google Scholar 

  38. W.S. Jewell, “Multi-commodity network solutions”, ORC-66-23, Operation Res. Center, Univ. of California (Berkeley) (Sept., 1966).

    Google Scholar 

  39. R. Saigal, “Multi-commodity flows in directed networks”, ORC 67-38, Univ. of California (Berkeley) (Sept., 1967).

    Google Scholar 

  40. M. Sakarovitch, “The multi-commodity maximal flow problem”, ORC 66-25, Univ. of California (Berkeley) (1966).

    Google Scholar 

  41. B. Rothschild and A. Whinston, “Feasibility of two commodity network flows”Operations Research 14 (6) (1966) 1121–1129.

    Google Scholar 

  42. M. Bellmore and J.C. Malone, “Pathology of traveling salesman algorithms”, Technical Report, The Johns Hopkins University.

  43. M. Bellmore and G.L. Nemhauser, “The traveling salesman problem: A survey”.Operations Research 16, No. 3 (1968) 538–558.

    Google Scholar 

  44. P.C. Gilmore and R.E. Gomory, “Sequencing a one-state variable machine: A solvable case of the traveling salesman problem”.Operations Research 12 (1964) 655–679.

    Google Scholar 

  45. M. Held and R.M. Karp, “The traveling-salesman problem and minimum spanning trees”,Operations Research to appear.

  46. T.C. Hu, “Decomposition of traveling salesman type problem”,Proc. IFORS, Session A, Theory of Graphs (1966) pp. A32–A44.

  47. S.L. Hakimi, “Optimal distribution of switching centers and medians of a graph”,Operations Research 12 (1964) 450–459.

    Google Scholar 

  48. E. Minieka, “Them-center problem”,SIAM Review 10, No. 1 (1970) 138–139.

    Google Scholar 

  49. J.B. Ackers, “The use of Wye-Delta transformation network simplification”,Operations Research 8, No. 3 (1960) 311–323.

    Google Scholar 

  50. A. Claus and D.J. Kleitman, “Heuristic methods for solving large scale network routing problems — The telepaking problem” (Tufts Univ. and MIT).

  51. E.J. Cockayne, “On the efficiency of the algorithm for Steiner minimal trees”,SIAM Journal of Applied Mathematics 18, No. 1 (1970) 150–159.

    Google Scholar 

  52. E.N. Gilbert and H.O. Pollak, “Steiner minimal trees”,Journal SIAM 16, No. 1 (1968) 1–29.

    Google Scholar 

  53. J.B. Kruskal, Jr., “On the shortest spanning trees of a graph and the traveling salesman problem”,Proceedings American Mathematical Society 7 (1956) 48–50.

    Google Scholar 

  54. R.C. Prim, “Shortest connection networks and some generalizations”,Bell System Technical Journal 36 (1957) 1389–1401.

    Google Scholar 

  55. B. Rothfarb and M. Goldstein, “One terminal telepak problem”, Office of Emergency Preparedness Report,Operations Research (to appear).

  56. T.C. Shah and R.D. Pederson, “Application of mathematical programming to optimal design of trunking networks”, Collins Radio Co., Dallas, Texas.

  57. V.N. Burkov, “A problem in decomposing graphs” (see ref. B22 of [A15]).

  58. M. Fujii, T. Kasami and K. Ninomiya, “Optimal sequencing of two equivalent processors”,SIAM Journal of Applied Mathematics 17 (4) (1969) 784–789.

    Google Scholar 

  59. T.C. Hu, “Parallel sequencing and assembly line problems”,Operations Research 9 (6) (1961) 841–848.

    Google Scholar 

  60. M.R. Garey, “Optimal binary decision trees for diagnostic identification problems”, Univ. of Wisconsin, Ph.D. Thesis in Computer Sciences, 1970.

  61. T.C. Hu and A.C. Tucker, “Optimal binary search trees”, MRC TSR #1049, Math. Res. Center, Univ. of Wisconsin, Madison, 1970.SIAM Journal of Applied Mathematics, to appear.

  62. D.E. Knuth, “Optimal binary search trees”, Computer Sciences Dept. Report 149, Stanford Univ., 1970.

  63. L.T. Reinwald and R.M. Soland, “Conversion of limited-entry decision tables to optimal computer programming” I:Journal of ACM 13 (1966) 339; II: 14 (1967) 742.

    Google Scholar 

  64. M. Balinkski, “Labelling to obtain a maximum matching”, in:Combinatorial mathematics and its applications, eds. Bose and Dowling, (Univ. of North Carolina Press, 1970).

  65. V.E. Benes,Mathematical theory of connecting networks and telephone traffic (Academic Press, New York, 1965).

    Google Scholar 

  66. R.W. Conway, W.L. Maxwell and L.W. Miller,Theory of scheduling (Addison-Wesley, New York, 1967). (See alsoNaval Research Logistics Quarterly 15, No. 2 (1968).)

    Google Scholar 

  67. R.J. Duffin, “Network models”, Dept. of Math., Carnegie-Mellon University Report 69-21, 1969.Proceedings of the Symposium on Mathematical Aspects of Electrical Network Theory (Am. Math. Soc., 1969).

  68. J. Edmonds, “Optimum branchings”,Mathematics of the Decision Sciences, Part 1, Lectures in Applied Math., Vol. 11 eds. G.B. Dantzig and A.F. Veinott, Jr. (Am. Math. Society, Providence, R.I., 1968).

    Google Scholar 

  69. S.L. Hakimi, “On the realization of a set of integers as degrees of the vertices of a linear graph”,SIAM Journal of Applied Mathematics 11 (1963) 135–147.

    Google Scholar 

  70. D.J. Kleitman, “Minimal number of multiple edges in realization of an incidence sequence without loops”,SIAM Journal of Applied Mathematics 18, No. 1 (1970) 25–28.

    Google Scholar 

  71. A.B. Owens, “On determining the minimum number of multiple edges for an incidence sequence”,SIAM Journal of Applied Mathematics 18, No. 1 (1970) 238–240.

    Google Scholar 

  72. P. Slepian,Mathematical foundations of networks (Springer-Verlag, Heidelberg, 1968).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Sponsored by the United States Army under Contract No.: DA-31-124-ARO-D-462 and the National Science Foundation, GJ-28339.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hu, T.C. Some problems in discrete optimization. Mathematical Programming 1, 102–112 (1971). https://doi.org/10.1007/BF01584076

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01584076

Keywords

Navigation