分区问题
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分区问题(英语:Partition problem)是数论和计算机科学领域的问题,[1]目的是把一个多重集分为和两个子集,要求和这两个集合中所有数的和相等。尽管分区问题属于NP完全问题,但是依然存在伪多项式时间的动态规划解法,而且在很多情况下也存在启发式的解法,能够求出最优解或近似最优解。正是基于这一点,这类问题也被称为“最简单的难题”。[2][3]
分区问题存在一个最佳化问题,该问题是将分为和,要求中元素的和与中元素的和相差最小。这一问题属于NP困难问题,但在实践中依旧存在高效的解法。[4]
分区问题是以下两个相关问题的特殊情况:
实例
现有多重集 ,可以被分为 以及 ,两者元素之和皆为5。
注释
- ^ Korf 1998.
- ^ Hayes, Brian, The Easiest Hard Problem (PDF), American Scientist, vol. 90 no. 2 (Sigma Xi, The Scientific Research Society), March–April 2002: 113–117 [2022-03-01], JSTOR 27857621, (原始内容存档 (PDF)于2012-09-16)
- ^ Mertens 2006,第125页.
- ^ Korf, Richard E. Multi-Way Number Partitioning (PDF). IJCAI. 2009 [2022-03-01]. (原始内容存档 (PDF)于2014-11-27).
- ^ Garey, Michael; Johnson, David. Computers and Intractability; A Guide to the Theory of NP-Completeness. 1979: 96–105. ISBN 978-0-7167-1045-5.
参考文献
- Borgs, Christian; Chayes, Jennifer; Pittel, Boris, Phase transition and finite-size scaling for the integer partitioning problem, Random Structures and Algorithms, 2001, 19 (3–4): 247–288, CiteSeerX 10.1.1.89.9577 , doi:10.1002/rsa.10004
- Gent, Ian; Walsh, Toby. Phase Transitions and Annealed Theories: Number Partitioning as a Case Study. Wolfgang Wahlster (编). Proceedings of 12th European Conference on Artificial Intelligence. ECAI-96. John Wiley and Sons: 170–174. August 1996. CiteSeerX 10.1.1.2.4475 .
- Gent, Ian; Walsh, Toby, Analysis of Heuristics for Number Partitioning, Computational Intelligence, 1998, 14 (3): 430–451, CiteSeerX 10.1.1.149.4980 , S2CID 15344203, doi:10.1111/0824-7935.00069
- Korf, Richard E., A complete anytime algorithm for number partitioning, Artificial Intelligence, 1998, 106 (2): 181–203, CiteSeerX 10.1.1.90.993 , ISSN 0004-3702, doi:10.1016/S0004-3702(98)00086-1
- Mertens, Stephan, Phase Transition in the Number Partitioning Problem, Physical Review Letters, November 1998, 81 (20): 4281–4284, Bibcode:1998PhRvL..81.4281M, S2CID 119541289, arXiv:cond-mat/9807077 , doi:10.1103/PhysRevLett.81.4281
- Mertens, Stephan, A physicist's approach to number partitioning, Theoretical Computer Science, 2001, 265 (1–2): 79–108, S2CID 16534837, arXiv:cond-mat/0009230 , doi:10.1016/S0304-3975(01)00153-0
- Mertens, Stephan. The Easiest Hard Problem: Number Partitioning. Allon Percus; Gabriel Istrate; Cristopher Moore (编). Computational complexity and statistical physics. USA: Oxford University Press. 2006: 125–140 [2022-03-01]. Bibcode:2003cond.mat.10317M. ISBN 9780195177374. arXiv:cond-mat/0310317 . (原始内容存档于2017-04-05).
- Mertens, Stephan, A complete anytime algorithm for balanced number partitioning, 1999, arXiv:cs/9903011