Reducibility Among Combinatorial Problems

Richard M. Karp, 1972, Complexity of Computer Computations

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组合问题之间的可归约性

Richard M. Karp

Richard M. Karp 的引言

整个20世纪60年代，我一直研究组合优化问题，包括与 Paul Roth 合作的逻辑电路设计、与 Mike Held 合作的装配线平衡问题和旅行商问题。这些经历使我意识到，看似简单的离散优化问题可能蕴含着组合爆炸的根源。Dantzig、Fulkerson、Hoffman、Edmonds、Lawler 以及其他先驱者在网络流、匹配和拟阵方面的工作，使我了解了有时可能实现的优雅而高效的算法。Jack Edmonds 的论文以及与他的几次关键讨论，使我注意到多项式时间可解性和超多项式时间可解性之间的关键区别。我还受到 Jack 对极小极大定理的强调所影响，他将其作为快速验证最优解的工具，而这一思想预示了 Steve Cook 对复杂度类 NP 的定义。另一个影响因素是 George Dantzig 提出的建议，即整数规划可以作为组合优化问题的通用形式。

整个60年代，我关注由 Rabin、Blum、Hartmanis、Stearns 以及其他人开创的计算复杂度理论的发展。在60年代末，我在布鲁克林理工学院教授一门关于该主题的课程时，研读了 Hartley Rogers 关于递归函数理论的精彩著作。这段经历使我深刻认识到可归约性在递归函数理论中的关键作用，并开始思考子递归可归约性是否能够在复杂度理论中发挥类似作用，但我当时还没有进一步研究这一类比。

Chapter 8 Reducibility Among Combinatorial Problems

Richard M. Karp

Introduction by 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.

Throughout the ’60s I followed developments in computational complexity theory, pioneered by Rabin, Blum, Hartmanis, Stearns and others. In the late ’60s I studied Hartley Rogers’ beautiful book on recursive function theory while teaching a course on the subject at the Polytechnic Institute of Brooklyn. This experience brought home to me the key role of reducibilities in recursive function theory, and started me wondering whether subrecursive reducibilities could play a similar role in complexity theory, but I did not yet pursue the analogy.

Cook’s 1971 paper [1], in which he defined the class NP and showed that propositional satisfiability was an NP-complete problem, brought together for me the two strands of complexity theory and combinatorial optimization. It was immediately apparent to me that many familiar combinatorial problems were likely to have the same universal role as satisfiability. I enjoyed constructing the polynomial-time reductions that verified this intuition. Most of them were easy to find, but I failed to prove the NP-completeness of the undirected hamiltonian circuit problem; that reduction was provided independently by Lawler and Tarjan. I was also frustrated by my inability to classify linear programming, graph isomorphism and primality.

As I recall I first presented my results at an informal seminar at Don Knuth’s home, and a few months later, in April 1972, I exposed the work more broadly at an IBM symposium. In the next couple of years many results more refined than my own were added to the accumulation of NP-completeness proofs, and later the Garey-Johnson book [2] presented the concepts to a wide audience.

Heuristic algorithms often find near-optimal solutions to NP-hard optimization problems. For some time in the mid-1970s I tried to explain this phenomenon by departing from worst-case analysis, and instead analyzing the performance of simple heuristics on instances drawn from simple probability distributions. This work was technically successful but gained limited traction, because there was no way to show that the problem instances drawn from these probability distributions are representative of those arising in practice. The surprising success of many heuristics remains a mystery.

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1 INTRODUCTION

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2 THE CLASS $P$

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3 NONDETERMINISTIC ALGORITHMS AND COOK'S THEOREM

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4 COMPLETE PROBLEMS

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5 NODE COVER

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6 SET COVERING

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7 FEEDBACK NODE SET

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8 FEEDBACK ARC SET

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9 DIRECTED HAMILTON CIRCUIT

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10 UNDIRECTED HAMILTON CIRCUIT

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11 SATISFIABILITY WITH AT MOST 3 LITERALS PER CLAUSE

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12 CHROMATIC NUMBER

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13 CLIQUE COVER

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14 EXACT COVER

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15 HITTING SET

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16 STEINER TREE

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17 3-DIMENSIONAL MATCHING

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18 KNAPSACK

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19 JOB SEQUENCING

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20 PARTITION

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21 MAX CUT

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Appendix I

References

  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. 2. M.R. Garey and D.S. Johnson, *Computers and Intractability: A Guide to the Theory of NP-Completeness*, W.H. Freeman & Co., 1979.
  2. 8 Reducibility Among Combinatorial Problems
  3. The following article originally appeared as:
  4. R.M. Karp, Reducibility Among Combinatorial Problems, Complexity of Computer Computations (R.E. Miller and J.W. Thatcher, eds.), Plenum Press, 1972, pp. 85–103.
  5. Copyright © 1972 Plenum Press.
  6. Reprinted by permission from Kluwer Academic Publishers.
  7. Richard M. Karp University of California at Berkeley
  8. Abstract: A large class of computational problems involve the determination of properties of graphs, digraphs, integers, arrays of integers, finite families of finite sets, boolean formulas and elements of other countable domains. Through simple encodings from such domains into the set of words over a finite alphabet these problems can be converted into language recognition problems, and we can inquire into their computational complexity. It is reasonable to consider such a problem satisfactorily solved when an algorithm for its solution is found which terminates within a number of steps bounded by a polynomial in the length of the input. We show that a large number of classic unsolved problems of covering, matching, packing, routing, assignment and sequencing are equivalent, in the sense that either each of them possesses a polynomial-bounded algorithm or none of them does.

Read off the page by a vision model reading the page images using olmOCR-2-7B-1025-FP8. Translated into 中文 by gpt-5, against glossary version 6. Translated from the English.

Read off the page by a vision model reading the page images using olmOCR-2-7B-1025-FP8.