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Reducibility Among Combinatorial Problems

In: 50 Years of Integer Programming 1958-2008

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  • Richard M. Karp

    (The University of California at Berkeley)

Abstract

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.

Suggested Citation

  • Richard M. Karp, 2010. "Reducibility Among Combinatorial Problems," Springer Books, in: Michael Jünger & Thomas M. Liebling & Denis Naddef & George L. Nemhauser & William R. Pulleyblank & (ed.), 50 Years of Integer Programming 1958-2008, chapter 0, pages 219-241, Springer.
    • Handle: RePEc:spr:sprchp:978-3-540-68279-0_8
    DOI: 10.1007/978-3-540-68279-0_8
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    Cited by:

    1. Shaposhnik, Yaron, 2025. "A dual-index rule for managing temporary congestion," European Journal of Operational Research, Elsevier, vol. 323(1), pages 34-44.
    2. Qian Hui & Tiandong Wang, 2025. "Systemic Risk Management via Maximum Independent Set in Extremal Dependence Networks," Papers 2503.15534, arXiv.org.
    3. Fan, Tianlong & Jiang, Wenjun & Zhang, Yi-Cheng & Lü, Linyuan, 2025. "A fast maximum clique algorithm based on network decomposition for large sparse networks," Chaos, Solitons & Fractals, Elsevier, vol. 194(C).

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