Some Estimated Likelihoods for Computational Complexity

Williams, Ryan

doi:10.1007/978-3-319-91908-9_2

Cited by 6 publications

(3 citation statements)

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“…We refer to a textbook such as[13] for an exposition of these circuit classes and corresponding lower bounds 2. Note that some researchers[16] do not see compelling evidence that the weak lower bounds required by Allender and Koucký[15] hold 3. The result is not stated in this way, but the proof shows that weak lower bounds are sufficient.…”

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confidence: 99%

Hardness Magnification for Natural Problems

Oliveira

Santhanam

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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confidence: 99%

Hardness Magnification for Natural Problems

Oliveira

Santhanam

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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“…For example, although SETH is based on the historically unsuccessful attempts on finding better algorithms for k-SAT, there is no consensus on its validity (see, e.g. [Wil16,Wil18b]).…”

Section: Weaker Complexity Assumptions For Approximation Hardnessmentioning

confidence: 99%

Fine-grained Complexity Meets IP = PSPACE

Chen¹,

Goldwasser²,

Lyu³

et al. 2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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In this paper we study the fine-grained complexity of finding exact and approximate solutions to problems in P. Our main contribution is showing reductions from an exact to an approximate solution for a host of such problems.As one (notable) example, we show that the Closest-LCS-Pair problem (Given two sets of strings A and B, compute exactly the maximum LCS(a, b) with (a, b) ∈ A × B) is equivalent to its approximation version (under near-linear time reductions, and with a constant approximation factor). More generally, we identify a class of problems, which we call BP-Pair-Class, comprising both exact and approximate solutions, and show that they are all equivalent under near-linear time reductions.Exploring this class and its properties, we also show:• Under the NC-SETH assumption (a significantly more relaxed assumption than SETH), solving any of the problems in this class requires essentially quadratic time.• Modest improvements on the running time of known algorithms (shaving log factors) would imply that NEXP is not in non-uniform NC 1 .• Finally, we leverage our techniques to show new barriers for deterministic approximation algorithms for LCS.A very important consequence of our results is that they continue to hold in the data structure setting. In particular, it shows that a data structure for approximate Nearest Neighbor Search for LCS (NNS LCS ) implies a data structure for exact NNS LCS and a data structure for answering regular expression queries with essentially the same complexity.At the heart of these new results is a deep connection between interactive proof systems for boundedspace computations and the fine-grained complexity of exact and approximate solutions to problems in P. In particular, our results build on the proof techniques from the classical IP = PSPACE result. * MIT, lijieche@mit.edu.The study of the fine-grained hardness of problems in P is one of the most interesting developments of the last few years in complexity theory. The study was initially aimed at the complexity of exact versions of important problem in P, such as Longest Common Subsequence (LCS), Edit Distance, All Pair Shortest Path (APSP), and 3-SUM. This was the natural starting point. There are several main thrusts of the study: establishing equivalence classes of problems that are "equivalent" to each other in the sense that a substantial improvement in one would imply a similar improvement in the other; showing fine-grained hardness under complexity assumptions, most notably the SETH1; and showing implications of even slight algorithmic improvements, such as "shaving-logs" off algorithms for P time problems, to circuit lower bounds.However, for many of these problems, approximate solutions are of interest as well, as they originate in natural problems which arise in pattern matching and bioinformatics [AVW14, BI15, BI16, BGL17, BK18], dynamic data structures [Pat10, AV14, AV14, HKNS15, KPP16, AD16, HLNV17, GKLP17], graph algorithms [RW13, GIKW17, AVY15, KT17], computational geometry [Bri14, Wil18a, DKL18, Che18] and ...

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“…The chapter "Some Estimated Likelihoods for Computational Complexity" by Williams [16] addresses some of the most fundamental open problems in computational complexity theory. Of course, we would like to know the answer to the P versus NP question (Is P = NP?).…”

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confidence: 99%