Computational Complexity: A Conceptual Perspective by Oded Goldreich

By Oded Goldreich

This booklet deals a complete viewpoint to trendy themes in complexity idea, that's a principal box of the theoretical foundations of laptop technological know-how. It addresses the looming query of what may be completed inside of a restricted period of time without or with different constrained typical computational assets. can be utilized as an creation for complex undergraduate and graduate scholars as both a textbook or for self-study, or to specialists, because it presents expositions of a number of the sub-areas of complexity conception resembling hardness amplification, pseudorandomness and probabilistic facts structures.

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Having mentioned the notion of approximation, we note that the study of the complexity of finding “approximate solutions” is also of natural importance. One type of approximation problems refers to an objective function defined on the set of potential solutions: Rather than finding a solution that attains the optimal value, the approximation task consists of finding a solution that attains an “almost optimal” value, where the notion of “almost optimal” may be understood in different ways giving rise to different levels of approximation.

Turning back to approximation problems, we note that in other cases a reasonable level of approximation is easier to achieve than solving the original (exact) search problem. Approximation is a natural relaxation of various computational problems. Another natural relaxation is the study of average-case complexity, where the “average” is taken over some “simple” distributions (representing a model of the problem’s instances that may occur in practice). We stress that, although it was not stated explicitly, the entire discussion so far has referred to “worst-case” analysis of algorithms.

Absolute Results (aka. Lower Bounds). As stated up-front, absolute results are not known for many of the “big questions” of Complexity Theory (most notably the P versus NP Question). However, several highly non-trivial absolute results have been proved. For example, it was shown that using negation can speed up the computation of monotone functions (which do not require negation for their mere computation). In addition, many promising techniques were introduced and employed with the aim of providing a low-level analysis of the progress of computation.

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