NP-Completeness

最新推荐文章于 2022-10-24 19:11:38 发布

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本文探讨了NP-完全性的核心概念，详细介绍了CIRCUIT-SAT问题作为首个被证明为NP-完全的问题，以及它如何通过多项式大小的电路来表达任何多项式时间内可验证的算法。文章深入分析了从算法到电路转换的过程，强调了输入位数固定的重要性，以及这种转换对于理解NP-完全性问题的关键作用。

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Theorem. CIRCUIT-SAT is NP-complete. [Cook 1971, Levin 1973]

Pf.(sketch)

Any algorithm that takes a fixed number of bits n as input and produces a yes/no answer can be represented by such a circuit. Moreover, if algorithm takes poly-time, then circuit if of poly-size.
sketchy part of proof; fixing the number of bits is important, and reflects basic distinction between algorithm and circuits.
Consider some problem X in NP. It has a poly-time certifier C(s,t). To determine whether s is in X, need to know if there exists a certificate t of length p(|s|) such that C(s,t) = yes.
View C(s,t) as an algorithm on |s| + p(|s|) bits(input s, certificate t) and convert it into a poly-size circuit.
- first |s| bits are hard-coded with s
- remaining p(|s|) bits are represent bits of t.
Circuit K is satisfiable iff C(s,t) = yes

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