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Computational Complexity of Combinatorial and Graph-Theoretic Problems

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Theoretical Computer Science

Part of the book series: C.I.M.E. Summer Schools ((CIME,volume 68))

Abstract

This series of lectures is concerned with efficient algorithms which operate on graphs. A graph is a structure consisting of a finite set of vertices, certain pairs of which are joined by edges. In an ordinary graph an edge has no direction specified. In a digraph each edge is directed from one vertex to another.

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References

  1. J. Beardwood, J.H. Halton and Ü.M. Hammersley, “The Shortest Path Through Many Points,” proc. Comb. Phil. Soc. 55 (1959), 299–327.

    Article  MathSciNet  MATH  Google Scholar 

  2. R. Bellman, “Combinatorial Processes and Dynamic Programming,” Proc. Tenth Symp. in Applied Math., American Mathematical Society (1960), 217–250.

    Google Scholar 

  3. C. Berge, “Two Theorems in Graph Theory,” Proa. Nat. Acad. Sci. 43 (1957), 842–844.

    Article  MathSciNet  MATH  Google Scholar 

  4. K.S. Booth and G.S. Lueker, “PQ-Tree Algorithms,” submitted to J. Comp, and Syst. Sci. (1975)

    Google Scholar 

  5. G.H. Bradley, “Equivalent Integer Programs and Canonical Problems,” Man. Sci. 17 (1971), 354–366.

    MathSciNet  MATH  Google Scholar 

  6. V. Chvátal, personal communication, 1973.

    Google Scholar 

  7. A. Cobham, “The Intrinsic Computational Difficulty of Functions,” Logic, Methodology and Philosophy of Science, North-Holland (1965).

    MATH  Google Scholar 

  8. S.A. Cook, “The Complexity of Theorem-Proving Procedures,” Proc. Third ACM Symp. on Theory of Computing (1971), 151–158.

    Google Scholar 

  9. S.A. Cook, “Characterizations of Pushdown Machines in Terms of Time- Bounded Computers,” J. Assoc. Comp. Mach. 18 (1971), 4–18.

    MATH  Google Scholar 

  10. S.Ar Cook and R.A. Reckhow, “Time Bounded Random Access Machines,” J. Cqmp. and Syst. Sei. 7 (1973), 354–375.

    Article  MathSciNet  MATH  Google Scholar 

  11. D. Corneil, see Proc. Fifth Southeastern Conference on Combinatorics3 Graph Theory and Computing, Utilitas Mathematica Publishing, Winnipeg (1974).

    Google Scholar 

  12. G.B. Dantzig, “On the Significance of Solving Linear Programming Problems with Some Integer Variables,” Econometrica 28 (1960), 30–44.

    Article  MathSciNet  MATH  Google Scholar 

  13. E. Dijkstra, “Note on Two Problems in Connection with Graphs,” Nurner. Math. 1 (1959), 269–271.

    Article  MathSciNet  MATH  Google Scholar 

  14. E.A. Dinic, “Algorithm for Solution of a Problem of Maximum Flow in a Network with Power Estimation,” Sov. Math. Doklad. 11 (1970), 1277–1280.

    Google Scholar 

  15. S.E. Dreyfus, “An Appraisal of Some Shortest Path Algorithms,” Operations Research 17 (1969), 395–412.

    Article  MATH  Google Scholar 

  16. J. Edmonds, “Paths, Trees and Flowers,” Canad. J. Math. 17 (1965), 449–467.

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Edmonds, “Minimum Partition of a Matroid into Independent Subsets, J. Res. NBS 69B (1965), 67–72.

    MathSciNet  Google Scholar 

  18. J. Edmonds and E.L. Johnson, “Matching: A Well-Solved Class of Integer Linear Programs,” Combinatorial Structures and Their Applications, Gordon and Breach (1970), 89–92.

    Google Scholar 

  19. J. Edmonds and R.M. Karp, “Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems,” J. Assoc. Comp. Mach. 9 (1972), 248–264.

    Google Scholar 

  20. P. Erdos and A. Renyi, “On Random Graphs, I,” Publications Mathematicae 6 (1959), 290–297.

    MATH  Google Scholar 

  21. P. Erdos and J. Spencer, Probabilistic Methods in Combinatorics, Academic Press (1974).

    MATH  Google Scholar 

  22. S. Even and o. Kariv, “An 0(n5/2) Algorithm for Maximum Matchings in General Graphs,” Proc. Sixteenth Annual Syrnp. on Foundations of Computer Science, IEEE (1975), 100–112.

    Google Scholar 

  23. R. Fagin, “Generalized First-Order Spectra and Polynomial-Time Recognizable Sets,” Complexity of Computation, ed. R. Karp, Proc.SIAM-AMS Symposia in Applied Mathematics (1974).

    Google Scholar 

  24. R.W. Floyd, “Nondeterministic Algorithms,” J. Assoc. Comp. Mach. 14 (1967), 636–644.

    MATH  Google Scholar 

  25. L.R. Ford and D.R. Fulkerson, Flows in Networks, Princeton University Press (1962).

    MATH  Google Scholar 

  26. H. Gabow, “An Efficient Implementation of Edmonds1 Maximum Matching Algorithm,” Technical Report No. 31, Stanford Digital Systems Laboratory (1972).

    Google Scholar 

  27. M.R. Garey, R.L. Graham and J.D. Ullman, “Worst-Case, Analysis of Memory Allocation Algorithms,” Proc. Fourth ACM Symposium on Theory of Computing (1972), 143–150.

    Google Scholar 

  28. M.R. Garey and D.S. Johnson, “The Efficiency of Near-Optimal Graph Coloring,” Bell Laboratories Report (1974).

    Google Scholar 

  29. M.R. Garey and D.S. Johnson, “Complexity Results for Multiprocessor Scheduling Under Resource Constraints,” Bell Telephone Laboratories (1974).

    Google Scholar 

  30. M.R. Garey, D.S. Johnson and L. Stockmeyef, “Some Simplified NP- Complete Graph Problems,” Proc. Sixth ACM Symposium on Theory of Computing(1974), 47–63; to appear in Theoretical Computer Science.

    Google Scholar 

  31. G.R. Grimmett and C.J.H. McDiarmid, “On Colouring Random Graphs,”Math. Proc. Comb. Phil. Soc. 77 (1975), 313–324.

    Article  MathSciNet  MATH  Google Scholar 

  32. J.E. Hopcroft and R.M. Karp, “An n5/2 Algorithm for Maximum Matchings in Bipartite Graphs,” SIAM J. Cornput. 2 (1973), 225–231.

    Article  MathSciNet  MATH  Google Scholar 

  33. J.E. Hopcroft and R.E. Tarjan, “Dividing a Graph Into Triconnected Components,” SIAM J. Comput. 2 (1973), 135–158.

    Article  MathSciNet  Google Scholar 

  34. J.E. Hopcroft and R.E. Tarjan, “Efficient Planarity Testing,” J. Assoc. Comput. Mach. 21 (1974), 549–568.

    MathSciNet  MATH  Google Scholar 

  35. J.E. Hopcroft and J.D. Ullman, Formal Languages and Their Relation tó Automata, Addison-Wesley, Reading, Mass. (1969).

    MATH  Google Scholar 

  36. O.H. Ibarra and C.E. Kim, “Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems,” J. Assoc. Comput. Mach. 22 (1975), 463–468.

    MathSciNet  MATH  Google Scholar 

  37. O.H. Ibarra and S. Sahni, “P-Complete Problems and Approximate Solutions,” Technical Report 74–5, University of Minnesota (1974).

    Google Scholar 

  38. D.S. Johnson, "Approximation Algorithms for Combinatorial Problems," to appear in J. Comp, and Syst. Sci.

    Google Scholar 

  39. D.S. Johnson, “Fast Algorithms for Bin Packing,” J. Comp. and Syst. Sci. 8 (1974), 272–314.

    Article  MATH  Google Scholar 

  40. D.S. Johnson, “Worst-Case Behavior of Graph Coloring Algorithms,”Proc. Fifth Southeastern Conference on Combinatorics 3 Graph Theory and Computing, Utilitas Mathematica Publishing, Winnipeg (1974).

    Google Scholar 

  41. R.M. Karp, “Reducibility Among Combinatorial Problems,” Complexity of Computer Computations, R.E. Miller and J.W. Thatcher, eds., Plenum Press, New York (1972), 85–104.

    Google Scholar 

  42. R.M. Karp, “On the Computational Complexity of Combinatorial Problems,” Networks 5 (1975), 45–68.

    MathSciNet  MATH  Google Scholar 

  43. R.M. Karp, A.C. McKellar and C.K. Wong, “Near-Optimal Solutions to a 2-Dimensional Placement Problem,” SIAM J. Comput., to appear in 1975.

    Google Scholar 

  44. V. Klee and G.J. Minty, “How Good is the Simplex Algorithm?,” Mathematical Note No. 643, Boeing Scientific Research Laboratories (1970).

    Google Scholar 

  45. D.E. Knuth, personal communication, 1974.

    Google Scholar 

  46. L. Kou, “Polynomial Complete Consecutive Information Retrieval Problems,” to appear in SIAM J. Comput.

    Google Scholar 

  47. R. Ladner, N. Lynch and A.L. Selman, “Comparison of Polynomial-Time Reducibilities,” Proc. Sixth ACM Symposium on Theory of Computing (1974), 110–121.

    Google Scholar 

  48. E.L. Lawler, Combinatorial Optimization: Networks and Matroids Holt, Rinehart and Winston, Inc., to appear.

    Google Scholar 

  49. E.L. Lawler, “Matroid Intersection Algorithms,” Mathematical Programming 9 (1975), 31–56.

    Article  MathSciNet  MATH  Google Scholar 

  50. S. Lin, “Heuristic Programming as an Aid to Network Design,” Proc. Symposium on Large-Scale Networks, Networks 5 (1975), 33–43.

    Google Scholar 

  51. D. Matula, “The Employee Party Problem,” abstract, Amer. Math. Soc. Notices (1972).

    Google Scholar 

  52. L. Posä, “Hamilton Circuits in Random Graphs,” to appear in Discrete Mathematics (1975).

    Google Scholar 

  53. R. Reiter, personal communication, 1971.

    Google Scholar 

  54. R. Rivest, personal communication, 1974.

    Google Scholar 

  55. D.J. Rose and R.E. Tarjan, “Algorithmic Aspects of Vertex Elimination,” Proc. Seventh Annual ACM Symposium on Theory of Computing (1975), 245–254.

    Google Scholar 

  56. D.J. Rosenkrantz, R.E. Stearns and P.M. Lewis, “Approximation Algorithms for the Traveling-Salesperson Problem,” Proc. Fifteenth IEEE Switching and Automata Theory Symposium (1974).

    Google Scholar 

  57. T. Schaefer, personal communication, 1974.

    Google Scholar 

  58. R. Tarjan, “Depth-First Search and Linear Graph Algorithms,” SIAM J, Comput. 1 (1972), 146–159.

    Article  MathSciNet  MATH  Google Scholar 

  59. R.E. Tarjan, “Finding Minimum Spanning Trees,” ERL-M501, Electronics Research Laboratory, University of California, Berkeley (1975).

    Google Scholar 

  60. J.E. Ullman, “Polynomial Complete Scheduling Problems,” Fourth Symp. on Operating System Principles (1973), 96–101.

    Google Scholar 

  61. A. Yao, “An 0(|E|log log|V|) Algorithm for Finding Minimum Spanning Trees,” to appear in Inf. Proc, Letters (1975).

    Google Scholar 

  62. N. Zadeh, “A Bad Network Problem for the Simplex Method and Other Minimum Cost Flow Algorithms,” Mathematical Programming 5 (1973), 255–266.

    Article  MathSciNet  MATH  Google Scholar 

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Karp, R.M. (2011). Computational Complexity of Combinatorial and Graph-Theoretic Problems. In: Preparata, F. (eds) Theoretical Computer Science. C.I.M.E. Summer Schools, vol 68. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11120-4_3

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