Algorithms and resource requirements for fundamental problems
Algorithms and resource requirements for fundamental problems
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基本问题的算法和资源需求
DOI：
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发表时间：
2007
期刊：
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通讯作者：
Ryan Williams
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文献类型：
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作者：
M. Blum;Ryan Williams
我们建立了更有效的方法来解决有趣的NP - 硬性问题，以及在负面方面可以解决这些问题和其他问题的限制。 ω（N2COS（π/7） - O（1））≥Ω（N1.801）通过任何使用（1）空间的算法求解的时间。 - 空间下限以达到满意（包括我们自己的），并且证明如何在我们的特定设置中进行自动化的搜索，而且还可以遵循某些高级模式的其他下限，我们描述了一个自动化的定理供体的实现，并强烈地提供了对上述时间的进一步改进，我们可以在较低的范围内进行大量的n sore n of new工具。详尽的搜索。对于所有可能的解决方案的详尽搜索，我们的算法在O（nδ）时间中解决了问题，对于通用常数Δ<0.792，这取决于我们还提供了一个较大的级别类型的Ald类型的较大级别的较大的级别的证据。
为了说明我们的结果，请考虑最大切割问题，其中给出了图G =（V，E）和整数K，并且希望确定G是否具有一定的顶点，使得离开子集的边缘的数量至少为K。显而易见的算法是在O（Poly（Poly（Poly（Poly（Poly））中，n no cotm no cotm copcience ye copcience ye copcience ye copcience ye contiment congiencations ye copcience ye contimence congiencation congiencations comportime comportim compitigity n = | | | | | | | | | | |工作。我们的结果表明，最大cut可以在O（3 n）时间求解，但不能在O（n1.801）时间和n o（1）空间中求解。
We establish more efficient methods for solving interesting classes of NP-hard problems exactly, as well as methods for proving limitations on how quickly those and other problems can be solved. (1) On the negative side, we prove that a number of NP-hard problems cannot be solved too efficiently by algorithms that only use a small amount of additional workspace. Building on prior work in the area, we prove that the Boolean satisfiability problem and other hard problems require Ω(n2cos(π/7)- o(1)) ≥ Ω( n1.801) time to solve by any algorithm that uses no(1) space. Stronger lower bounds are proved for solving quantified Boolean formulas with a fixed number of quantifiers. Our results are essentially model-independent, in that they hold for all reasonable random-access machine models. Furthermore, we present a formal proof system that captures all prior time-space lower bounds for satisfiability (including our own), and demonstrate how the search for better lower bounds can be automated, in not only our particular setting but also other lower bounds that follow a certain high-level pattern. We describe an implementation of an automated theorem prover and provide experimental results which strongly suggest that further improvements on the above time lower bound will require new tools and ideas. (2) On the positive side, we give a general methodology for solving a large class of NP-hard problems much faster than exhaustive search. In particular, for a problem in the class where exhaustive search of all possible solutions takes Θ(N ) time, our algorithm solves the problem in O( Nδ) time, for a universal constant δ < 0.792 that depends on the complexity of multiplying two matrices over a ring. We also provide theoretical evidence that a much larger class of problems admits a similar type of algorithm.
To illustrate our results, consider the MAX CUT problem, where one is given a graph G = ( V, E) and integer K, and one wishes to determine if G has a subset of vertices such that the number of edges leaving the subset is at least K. The obvious algorithm for MAX CUT runs in O(poly( n) · 2n) time, where n = |V|. Despite the problem's importance, no better algorithm was known for the general case of MAX CUT , prior to our work. Our results imply that MAX C UT can be solved in O( 3 n ) time but cannot be solved in O(n1.801) time and n o(1) space.