The Landscape of Communication Complexity Classes

Göös, Mika; Pitassi, Toniann; Watson, Thomas

doi:10.1007/s00037-018-0166-6

Cited by 36 publications

(37 citation statements)

References 72 publications

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“…We remark that there are no non-trivial lower bounds known against Σ cc 2 protocols [GPW18], which suggests that Σ cc 2 protocols could be very powerful, and the first approach (Theorem 1.14) may be applicable to many other problems. This is not the case for MA cc protocols which were used in several previous works [ARW17, Rub18, KLM18, Che18]: for example, there is an essentially tight Ω( √ n) MA cc lower bound for Set-Disjointness [Kla03, AW09, Che18].…”

Section: An Equivalence Class For Sparse Orthogonal Vectorsmentioning

confidence: 99%

An Equivalence Class for Orthogonal Vectors

Chen

Williams

2019

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

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The Orthogonal Vectors problem (OV) asks: given n vectors in {0, 1} O(log n) , are two of them orthogonal? OV is easily solved in O(n 2 log n) time, and it is a central problem in fine-grained complexity: dozens of conditional lower bounds are based on the popular hypothesis that OV cannot be solved in (say) n 1.99 time. However, unlike the APSP problem, few other problems are known to be non-trivially equivalent to OV.We show OV is truly-subquadratic equivalent to several fundamental problems, all of which (a priori) look harder than OV. A partial list is given below:

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Section: An Equivalence Class For Sparse Orthogonal Vectorsmentioning

confidence: 99%

An Equivalence Class for Orthogonal Vectors

Chen

Williams

2019

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

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“…It is worth noting that IP communication lower bounds are extremely hard to prove-proving a non-trivial lower bound for AM communication protocols is already a long-standing open question [Lok01, GPW16,GPW18]. Hence, resolving Open Question 3 negatively could be hard.…”

Section: Open Question 3 Is There An O(log D) Time Ip Communication mentioning

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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“…Very recently, Bouland et al [9] have "explained" our inability to prove lower bounds on OIP [2] + + + protocols: they showed that the OIP [2] + + + model, as well as 2-message SIPs themselves, are powerful enough to compute (partial) functions outside of UPP. UPP is the most powerful two-party communication model against which existing methods can prove lower bounds (see, e.g., [22]). Hence, proving OIP [2] + + + lower bounds is likely to require substantially new techniques.…”

Section: Related Workmentioning

confidence: 99%

Verifiable Stream Computation and Arthur--Merlin Communication

Chakrabarti¹,

Cormode²,

McGregor³

et al. 2019

SIAM J. Comput.

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In the setting of streaming interactive proofs (SIPs), a client (verifier) needs to compute a given function on a massive stream of data, arriving online, but is unable to store even a small fraction of the data. It outsources the processing to a third party service (prover), but is unwilling to blindly trust answers returned by this service. Thus, the service cannot simply supply the desired answer; it must convince the verifier of its correctness via a short interaction after the stream has been seen. In this work we study "barely interactive" SIPs. Specifically, we show that one or two rounds of interaction suffice to solve several query problems-including Index, Median, Nearest Neighbor Search, Pattern Matching, and Range Counting-with polylogarithmic space and communication costs. Such efficiency with O(1) rounds of interaction was thought to be impossible based on previous work. On the other hand, we initiate a formal study of the limitations of constant-round SIPs by introducing a new hierarchy of communication models called Online Interactive Proofs (OIPs). The online nature of these models is analogous to the streaming restriction placed upon the verifier in a SIP. We give upper and lower bounds that (1) characterize, up to quadratic blowups, every finite level of the OIP hierarchy in terms of other well-known communication complexity classes, (2) separate the first four levels of the hierarchy, and (3) reveal that the hierarchy collapses to the fourth level. Our study of OIPs reveals marked contrasts and some parallels with the classic Turing Machine theory of interactive proofs, establishes limits on the power of existing techniques for developing constantround SIPs, and provides a new characterization of (non-online) Arthur-Merlin communication in terms of an online model.

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The Landscape of Communication Complexity Classes

Cited by 36 publications

References 72 publications

An Equivalence Class for Orthogonal Vectors

An Equivalence Class for Orthogonal Vectors

Fine-grained Complexity Meets IP = PSPACE

Verifiable Stream Computation and Arthur--Merlin Communication

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