Reducibility Among Combinatorial Problems

  • Richard M. Karp


Throughout the 1960s I worked on combinatorial optimization problems including logic circuit design with Paul Roth and assembly line balancing and the traveling salesman problem with Mike Held. These experiences made me aware that seemingly simple discrete optimization problems could hold the seeds of combinatorial explosions. The work of Dantzig, Fulkerson, Hoffman, Edmonds, Lawler and other pioneers on network flows, matching and matroids acquainted me with the elegant and efficient algorithms that were sometimes possible. Jack Edmonds’ papers and a few key discussions with him drew my attention to the crucial distinction between polynomial-time and superpolynomial-time solvability. I was also influenced by Jack’s emphasis on min-max theorems as a tool for fast verification of optimal solutions, which foreshadowed Steve Cook’s definition of the complexity class NP. Another influence was George Dantzig’s suggestion that integer programming could serve as a universal format for combinatorial optimization problems.


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  1. 1.
    S.A. Cook, The complexity of theorem proving procedures, Proceedings of the 3rd Annual ACM Symposium on Theory of Computing, 1971, pp. 151–158.Google Scholar
  2. 2.
    M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NPCompleteness, W.H. Freeman & Co., 1979.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.The University of California at BerkeleyBerkeleyUSA

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