Zero-Information Protocols and Unambiguity in Arthur-Merlin Communication

Göös, Mika; Pitassi, Toniann; Watson, Thomas

doi:10.1145/2688073.2688074

Cited by 12 publications

(24 citation statements)

References 60 publications

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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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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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“…This model, which is well studied in the communication complexity literature [BFS86,Kla03,Kla10], is viewed as the communication complexity analogue of AM protocols [BM88]. Recently, Göös, Pitassi and Watson [GPW15] suggested a variant of this model which requires an additional "zero-knowledge" property defined as follows: For any 1-input (x, y) ∈ f −1 (1), the proof sent by the honest prover provides no information on the inputs (x, y) to an external viewer. Formally, the random variable p x,y = p(x, y, r A , r B ) induced by a random choice of r A and r B should be distributed according to some universal distribution D which is independent of the specific 1-input (x, y).…”

Section: Zero-information Unambiguous Arthur-merlin Communication Promentioning

confidence: 99%

“…The picture becomes even more confusing when looking at existing constructions. On one hand, the general ZAM constructions presented by [GPW15,Theorem 6] (which use a reduction to Disjointness) seem more elementary than the simplest PSM protocols of [FKN94]. On the other hand, there are ZAM constructions which share common ingredients with existing PSM protocols.…”

Section: Zam Vs Psmmentioning

confidence: 99%

From Private Simultaneous Messages to Zero-Information Arthur–Merlin Protocols and Back

Applebaum

Raykov

2016

J Cryptol

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Göös, Pitassi and Watson (ITCS, 2015) have recently introduced the notion of Zero-Information Arthur-Merlin Protocols (ZAM). In this model, which can be viewed as a private version of the standard Arthur-Merlin communication complexity game, Alice and Bob are holding a pair of inputs x and y respectively, and Merlin, the prover, attempts to convince them that some public function f evaluates to 1 on (x, y). In addition to standard completeness and soundness, Göös et al., require a "zero-knowledge" property which asserts that on each yes-input, the distribution of Merlin's proof leaks no information about the inputs (x, y) to an external observer. In this paper, we relate this new notion to the well-studied model of Private Simultaneous Messages (PSM) that was originally suggested by Feige, Naor and Kilian (STOC, 1994). Roughly speaking, we show that the randomness complexity of ZAM corresponds to the communication complexity of PSM, and that the communication complexity of ZAM corresponds to the randomness complexity of PSM. This relation works in both directions where different variants of PSM are being used. As a secondary contribution, we reveal new connections between different variants of PSM protocols which we believe to be of independent interest. Our results give rise to better ZAM protocols based on existing PSM protocols, and to better protocols for conditional disclosure of secrets (a variant of PSM) from existing ZAMs.

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“…Theorem 3 has consequences for Arthur-Merlin communication (MA cc , AM cc ) which has been studied in [26,40,2,15,29,19,17]. Namely, Klauck [26] asked (using the language of uniform threshold covers) whether the known inclusion MA cc ⊆ SBP cc is strict.…”

Section: Theorem 3 Sbp CC Is Not Closed Under Intersectionmentioning

confidence: 99%

“…Motivated by this [17] studied a certain unambiguous restriction of AM cc , denoted UAM cc , as a stepping stone towards AM cc . They asked whether UAM cc ⊆ SBP cc .…”

Section: Theorem 3 Sbp CC Is Not Closed Under Intersectionmentioning

confidence: 99%

Rectangles Are Nonnegative Juntas

Göös

Lovett

Meka

et al. 2015

Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing

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We develop a new method to prove communication lower bounds for composed functions of the form f • g n where f is any boolean function on n inputs and g is a sufficiently "hard" two-party gadget. Our main structure theorem states that each rectangle in the communication matrix of f • g n can be simulated by a nonnegative combination of juntas. This is the strongest yet formalization for the intuition that each low-communication randomized protocol can only "query" few inputs of f as encoded by the gadget g. Consequently, we characterize the communication complexity of f • g n in all known one-sided zero-communication models by a corresponding query complexity measure of f . These models in turn capture important lower bound techniques such as corruption, smooth rectangle bound, relaxed partition bound, and extended discrepancy.As applications, we resolve several open problems from prior work: We show that SBP cc (a class characterized by corruption) is not closed under intersection. An immediate corollary is that MA cc = SBP cc . These results answer questions of Klauck (CCC 2003) and Böhler et al. (JCSS 2006). We also show that approximate nonnegative rank of partial boolean matrices does not admit efficient error reduction. This answers a question of Kol et al. (ICALP 2014) for partial matrices.

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Zero-Information Protocols and Unambiguity in Arthur-Merlin Communication

Cited by 12 publications

References 60 publications

Fine-grained Complexity Meets IP = PSPACE

Fine-grained Complexity Meets IP = PSPACE

From Private Simultaneous Messages to Zero-Information Arthur–Merlin Protocols and Back

Rectangles Are Nonnegative Juntas

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