Showing papers on "Communication complexity published in 2015"

Codes for distributed PIR with low storage overhead

Abstract: Private information retrieval (PIR) protocols allow a user to retrieve a data item from a database without revealing any information about the identity of the item being retrieved. Specifically, in information-theoretic k-server PIR, the database is replicated among k non-communicating servers, and each server learns nothing about the item retrieved by the user. The cost of PIR protocols is usually measured in terms of their communication complexity, which is the total number of bits exchanged between the user and the servers. However, another important cost parameter is the storage overhead, which is the ratio between the total number of bits stored on all the servers and the number of bits in the database. Since single-server information-theoretic PIR is impossible, the storage overhead of all existing PIR protocols is at least 2 (or k, in the case of k-server PIR). In this work, we show that information-theoretic PIR can be achieved with storage overhead arbitrarily close to the optimal value of 1, without sacrificing the communication complexity. Specifically, we prove that all known k-server PIR protocols can be efficiently emulated, while preserving both privacy and communication complexity but significantly reducing the storage overhead. To this end, we distribute the n bits of the database among s + r servers, each storing n/s coded bits (rather than replicas). Notably, our coding scheme remains the same, regardless of the specific k-server PIR protocol being emulated. For every fixed k, the resulting storage overhead (s +r)/s approaches 1 as s grows; explicitly we have equation. Moreover, in the special case k = 2, the storage overhead is only 1 + 1/s. In order to achieve these results, we introduce and study a new kind of binary linear codes, called here k-server PIR codes. Finally, we show how such codes can be constructed from multidimensional cubic, from Steiner systems, and from one-step majority-logic decodable codes.

Distributed Caching for Data Dissemination in the Downlink of Heterogeneous Networks

Abstract: Heterogeneous cellular networks (HCNs) with embedded small cells are considered, where multiple mobile users wish to download network content of different popularity. By caching data into the small-cell base stations, we will design distributed caching optimization algorithms via belief propagation (BP) for minimizing the downloading latency. First, we derive the delay-minimization objective function and formulate an optimization problem. Then, we develop a framework for modeling the underlying HCN topology with the aid of a factor graph. Furthermore, a distributed BP algorithm is proposed based on the network's factor graph. Next, we prove that a fixed point of convergence exists for our distributed BP algorithm. In order to reduce the complexity of the BP, we propose a heuristic BP algorithm. Furthermore, we evaluate the average downloading performance of our HCN for different numbers and locations of the base stations and mobile users, with the aid of stochastic geometry theory. By modeling the nodes distributions using a Poisson point process, we develop the expressions of the average factor graph degree distribution, as well as an upper bound of the outage probability for random caching schemes. We also improve the performance of random caching. Our simulations show that 1) the proposed distributed BP algorithm has a near-optimal delay performance, approaching that of the high-complexity exhaustive search method; 2) the modified BP offers a good delay performance at low communication complexity; 3) both the average degree distribution and the outage upper bound analysis relying on stochastic geometry match well with our Monte-Carlo simulations; and 4) the optimization based on the upper bound provides both a better outage and a better delay performance than the benchmarks.

Quantum superposition of the order of parties as a communication resource

Abstract: In a variant of communication complexity tasks, two or more separated parties cooperate to compute a function of their local data, using a limited amount of communication. It is known that communication of quantum systems and shared entanglement can increase the probability for the parties to arrive at the correct value of the function, compared to classical resources. Here we show that quantum superpositions of the direction of communication between parties can also serve as a resource to improve the probability of success. We present a tripartite task for which such a superposition provides an advantage compared to the case where the parties communicate in a fixed order. In a more general context, our result also provides semi-device-independent certification of the absence of a definite order of communication.

Communication complexity of distributed convex learning and optimization

Abstract: We study the fundamental limits to communication-efficient distributed methods for convex learning and optimization, under different assumptions on the information available to individual machines, and the types of functions considered. We identify cases where existing algorithms are already worst-case optimal, as well as cases where room for further improvement is still possible. Among other things, our results indicate that without similarity between the local objective functions (due to statistical data similarity or otherwise) many communication rounds may be required, even if the machines have unbounded computational power.

On the Communication Complexity of Secure Function Evaluation with Long Output

Abstract: We study the communication complexity of secure function evaluation (SFE). Consider a setting where Alice has a short input χA, Bob has an input χB and we want Bob to learn some function y = f(χA, χB) with large output size. For example, Alice has a small secret decryption key, Bob has a large encrypted database and we want Bob to learn the decrypted data without learning anything else about Alice's key. In a trivial insecure protocol, Alice can just send her short input χA to Bob. However, all known SFE protocols have communication complexity that scales with size of the output y, which can potentially be much larger. Is such 'output-size dependence' inherent in SFE'Surprisingly, we show that output-size dependence can be avoided in the honest-but-curious setting. In particular, using indistinguishability obfuscation (iO) and fully homomorphic encryption (FHE), we construct the first honest-but-curious SFE protocol whose communication complexity only scales with that of the best insecure protocol for evaluating the desired function, independent of the output size. Our construction relies on a novel way of using iO via a new tool that we call a 'somewhere statistically binding (SSB) hash', and which may be of independent interest.On the negative side, we show that output-size dependence is inherent in the fully malicious setting, or even already in an honest-but-deterministic setting, where the corrupted party follows the protocol as specified but fixes its random tape to some deterministic value. Moreover, we show that even in an offline/online protocol, the communication of the online phase must have output-size dependence. This negative result uses an incompressibility argument and it generalizes several recent lower bounds for functional encryption and (reusable) garbled circuits, which follow as simple corollaries of our general theorem.

GRECS: Graph Encryption for Approximate Shortest Distance Queries

Abstract: We propose graph encryption schemes that efficiently support approximate shortest distance queries on large-scale encrypted graphs. Shortest distance queries are one of the most fundamental graph operations and have a wide range of applications. Using such graph encryption schemes, a client can outsource large-scale privacy-sensitive graphs to an untrusted server without losing the ability to query it. Other applications include encrypted graph databases and controlled disclosure systems. We propose GRECS (stands for GRaph EnCryption for approximate Shortest distance queries) which includes three oracle encryption schemes that are provably secure against any semi-honest server. Our first construction makes use of only symmetric-key operations, resulting in a computationally-efficient construction. Our second scheme makes use of somewhat-homomorphic encryption and is less computationally-efficient but achieves optimal communication complexity (i.e. uses a minimal amount of bandwidth). Finally, our third scheme is both computationally-efficient and achieves optimal communication complexity at the cost of a small amount of additional leakage. We implemented and evaluated the efficiency of our constructions experimentally. The experiments demonstrate that our schemes are efficient and can be applied to graphs that scale up to 1.6 million nodes and 11 million edges.

GRECS: Graph Encryption for Approximate Shortest Distance Queries.

Abstract: We propose graph encryption schemes that efficiently support approximate shortest distance queries on large-scale encrypted graphs. Shortest distance queries are one of the most fundamental graph operations and have a wide range of applications. Using such graph encryption schemes, a client can outsource large-scale privacy-sensitive graphs to an untrusted server without losing the ability to query it. Other applications include encrypted graph databases and controlled disclosure systems. We propose GRECS (stands for GRaph EnCryption for approximate Shortest distance queries) which includes three schemes that are provably secure against any semi-honest server. Our first construction makes use of only symmetric-key operations, resulting in a computationally-efficient construction. Our second scheme, makes use of somewhat-homomorphic encryption and is less computationally-efficient but achieves optimal communication complexity (i.e., uses a minimal amount of bandwidth). Finally, our third scheme is both computationally-efficient and achieves optimal communication complexity at the cost of a small amount of additional leakage. We implemented and evaluated the efficiency of our constructions experimentally. The experiments demonstrate that our schemes are efficient and can be applied to graphs that scale up to 1.6 million nodes and 11 million edges.

PIR with Low Storage Overhead: Coding instead of Replication

Abstract: Private information retrieval (PIR) protocols allow a user to retrieve a data item from a database without revealing any information about the identity of the item being retrieved. Specifically, in information-theoretic $k$-server PIR, the database is replicated among $k$ non-communicating servers, and each server learns nothing about the item retrieved by the user. The cost of PIR protocols is usually measured in terms of their communication complexity, which is the total number of bits exchanged between the user and the servers, and storage overhead, which is the ratio between the total number of bits stored on all the servers and the number of bits in the database. Since single-server information-theoretic PIR is impossible, the storage overhead of all existing PIR protocols is at least $2$. In this work, we show that information-theoretic PIR can be achieved with storage overhead arbitrarily close to the optimal value of $1$, without sacrificing the communication complexity. Specifically, we prove that all known $k$-server PIR protocols can be efficiently emulated, while preserving both privacy and communication complexity but significantly reducing the storage overhead. To this end, we distribute the $n$ bits of the database among $s+r$ servers, each storing $n/s$ coded bits (rather than replicas). For every fixed $k$, the resulting storage overhead $(s+r)/s$ approaches $1$ as $s$ grows; explicitly we have $r\le k\sqrt{s}(1+o(1))$. Moreover, in the special case $k = 2$, the storage overhead is only $1 + \frac{1}{s}$. In order to achieve these results, we introduce and study a new kind of binary linear codes, called here $k$-server PIR codes. We then show how such codes can be constructed, and we establish several bounds on the parameters of $k$-server PIR codes. Finally, we briefly discuss extensions of our results to nonbinary alphabets, to robust PIR, and to $t$-private PIR.

Deterministic Communication vs. Partition Number.

Abstract: We show that deterministic communication complexity can be superlogarithmic in the partition number of the associated communication matrix. We also obtain near-optimal deterministic lower bounds fo...

Lower Bounds on Information Complexity via Zero-Communication Protocols and Applications

Abstract: We show that almost all known lower bound methods for communication complexity are also lower bounds for the information complexity. In particular, we define a relaxed version of the partition bound of Jain and Klauck [Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity, 2010, pp. 247--258] and prove that it lower bounds the information complexity of any function. Our relaxed partition bound subsumes all norm-based methods (e.g., the $\gamma_2$ method) and rectangle-based methods (e.g., the rectangle/corruption bound, the smooth rectangle bound, and the discrepancy bound), except the partition bound. Our result uses a new connection between rectangles and zero-communication protocols, where the players can either output a value or abort. We prove, using a sampling protocol designed by Braverman and Weinstein [in Approximation, Randomization, and Combinatorial Optimization, Lecture Notes in Comput. Sci. 7408, Springer, Heidelberg, 2012, pp. 459--470], the following compression l...

Deterministic Communication vs. Partition Number

Abstract: We show that deterministic communication complexity can be super logarithmic in the partition number of the associated communication matrix. We also obtain near-optimal deterministic lower bounds for the Clique vs. Independent Set problem, which in particular yields new lower bounds for the log-rank conjecture. All these results follow from a simple adaptation of a communication-to-query simulation theorem of Raz and McKenzie (Combinatorica 1999) together with lower bounds for the analogous query complexity questions.

Distributed Block Coordinate Descent for Minimizing Partially Separable Functions

Abstract: A distributed randomized block coordinate descent method for minimizing a convex function of a huge number of variables is proposed. The complexity of the method is analyzed under the assumption that the smooth part of the objective function is partially block separable. The number of iterations required is bounded by a function of the error and the degree of separability, which extends the results in Richtarik and Takac (Parallel Coordinate Descent Methods for Big Data Optimization, Mathematical Programming, DOI:10.1007/s10107-015-0901-6) to a distributed environment. Several approaches to the distribution and synchronization of the computation across a cluster of multi-core computer are described and promising computational results are provided.

Adaptive Distributed Outlier Detection for WSNs

Abstract: The paradigm of pervasive computing is gaining more and more attention nowadays, thanks to the possibility of obtaining precise and continuous monitoring. Ease of deployment and adaptivity are typically implemented by adopting autonomous and cooperative sensory devices; however, for such systems to be of any practical use, reliability and fault tolerance must be guaranteed, for instance by detecting corrupted readings amidst the huge amount of gathered sensory data. This paper proposes an adaptive distributed Bayesian approach for detecting outliers in data collected by a wireless sensor network; our algorithm aims at optimizing classification accuracy, time complexity and communication complexity, and also considering externally imposed constraints on such conflicting goals. The performed experimental evaluation showed that our approach is able to improve the considered metrics for latency and energy consumption, with limited impact on classification accuracy.

A Short Proof that the Extension Complexity of the Correlation Polytope Grows Exponentially

Abstract: We establish that the extension complexity of the $$n\times n$$n×n correlation polytope is at least $$1.5\,^n$$1.5n by a short proof that is self-contained except for using the fact that every face of a polyhedron is the intersection of all facets it is contained in. The main innovative aspect of the proof is a simple combinatorial argument showing that the rectangle covering number of the unique-disjointness matrix is at least $$1.5^n$$1.5n, and thus the nondeterministic communication complexity of the unique-disjointness predicate is at least $$.58n$$.58n. We thereby slightly improve on the previously best known lower bounds $$1.24^n$$1.24n and $$.31n$$.31n, respectively.

Constant Communication ORAM with Small Blocksize

Abstract: There have been several attempts recently at using homomorphic encryption to increase the efficiency of Oblivious RAM protocols. One of the most successful has been Onion ORAM, which achieves O(1) communication overhead with polylogarithmic server computation. However, it has two drawbacks. It requires a large block size of B = Ω(log6 N) with large constants. Moreover, while it only needs polylogarithmic computation complexity, that computation consists mostly of expensive homomorphic multiplications. In this work, we address these problems and reduce the required block size to Ω(log4 N). We remove most of the homomorphic multiplications while maintaining O(1) communication complexity. Our idea is to replace their homomorphic eviction routine with a new, much cheaper permute-and-merge eviction which eliminates homomorphic multiplications and maintains the same level of security. In turn, this removes the need for layered encryption that Onion ORAM relies on and reduces both the minimum block size and server computation.

Secure Comparison Protocols in the Semi-Honest Model

Abstract: Due to high complexity, comparison protocols with secret inputs have been a bottleneck in the design of privacy-preserving cryptographic protocols. Different solutions based on homomorphic encryption, garbled circuits and secret sharing techniques have been proposed over the last few years, each claiming high efficiency. Unfortunately, a fair comparison of existing protocols in terms of run-time, bandwidth requirement and round complexity has been lacking so far. In this paper, we analyze the state-of-the-art comparison protocols for a two-party setting in the semi-honest security protocol. We analyze their performances in three stages, namely initialization, pre-processing and online computation, by implementing them on a single platform. The results of our experiments provide a clear insight for the research community into the advantages and disadvantages of the various techniques .

Extended formulations, nonnegative factorizations, and randomized communication protocols

Abstract: An extended formulation of a polyhedron $$P$$P is a linear description of a polyhedron $$Q$$Q together with a linear map $$\pi $$? such that $$\pi (Q)=P$$?(Q)=P. These objects are of fundamental importance in polyhedral combinatorics and optimization theory, and the subject of a number of studies. Yannakakis' factorization theorem (Yannakakis in J Comput Syst Sci 43(3):441---466, 1991) provides a surprising connection between extended formulations and communication complexity, showing that the smallest size of an extended formulation of $$P$$P equals the nonnegative rank of its slack matrix $$S$$S. Moreover, Yannakakis also shows that the nonnegative rank of $$S$$S is at most $$2^c$$2c, where $$c$$c is the complexity of any deterministic protocol computing $$S$$S. In this paper, we show that the latter result can be strengthened when we allow protocols to be randomized. In particular, we prove that the base-$$2$$2 logarithm of the nonnegative rank of any nonnegative matrix equals the minimum complexity of a randomized communication protocol computing the matrix in expectation. Using Yannakakis' factorization theorem, this implies that the base-$$2$$2 logarithm of the smallest size of an extended formulation of a polytope $$P$$P equals the minimum complexity of a randomized communication protocol computing the slack matrix of $$P$$P in expectation. We show that allowing randomization in the protocol can be crucial for obtaining small extended formulations. Specifically, we prove that for the spanning tree and perfect matching polytopes, small variance in the protocol forces large size in the extended formulation.

Communication Complexity of Conditional Disclosure of Secrets and Attribute-Based Encryption

Abstract: We initiate a systematic treatment of the communication complexity of conditional disclosure of secrets (CDS), where two parties want to disclose a secret to a third party if and only if their respective inputs satisfy some predicate. We present a general upper bound and the first non-trivial lower bounds for conditional disclosure of secrets. Moreover, we achieve tight lower bounds for many interesting setting of parameters for CDS with linear reconstruction, the latter being a requirement in the application to attribute-based encryption. In particular, our lower bounds explain the trade-off between ciphertext and secret key sizes of several existing attribute-based encryption schemes based on the dual system methodology.

Distributed Stochastic Variance Reduced Gradient Methods and A Lower Bound for Communication Complexity

Abstract: We study distributed optimization algorithms for minimizing the average of convex functions. The applications include empirical risk minimization problems in statistical machine learning where the datasets are large and have to be stored on different machines. We design a distributed stochastic variance reduced gradient algorithm that, under certain conditions on the condition number, simultaneously achieves the optimal parallel runtime, amount of communication and rounds of communication among all distributed first-order methods up to constant factors. Our method and its accelerated extension also outperform existing distributed algorithms in terms of the rounds of communication as long as the condition number is not too large compared to the size of data in each machine. We also prove a lower bound for the number of rounds of communication for a broad class of distributed first-order methods including the proposed algorithms in this paper. We show that our accelerated distributed stochastic variance reduced gradient algorithm achieves this lower bound so that it uses the fewest rounds of communication among all distributed first-order algorithms.

Randomized Communication vs. Partition Number.

Abstract: We show that randomized communication complexity can be superlogarithmic in the partition number of the associated communication matrix, and we obtain near-optimal randomized lower bounds for the Clique vs. Independent Set problem. These results strengthen the deterministic lower bounds obtained in prior work (Goos, Pitassi, and Watson, FOCS 2015). One of our main technical contributions states that information complexity when the cost is measured with respect to only 1-inputs (or only 0-inputs) is essentially equivalent to information complexity with respect to all inputs.

Perfect Quantum Network Coding Independent of Classical Network Solutions

Abstract: Quantum network coding with classical assistance can transmit quantum information with fidelity one in error-free quantum channels. However, the dependency on classical network solutions leads to higher communication complexity and transmission congestion over a large-scale network with bottleneck channels. We find that certain types of quantum operation are sufficient to support multiple unicast sessions over the extended butterfly network. Furthermore, this construction decouples classical communication (assisting decoding information) from classical network solution. Based on this design, we propose an optimal protocol for this specific network, which significantly reduces the communication cost.

Quantum Private Information Retrieval has Linear Communication Complexity

Abstract: In private information retrieval (PIR), a client queries an $$n$$ n -bit database in order to retrieve an entry of her choice, while maintaining privacy of her query value. Chor et al. [J ACM 45(6):965---981, 1998] showed that, in the information-theoretical setting, a linear amount of communication is required for classical PIR protocols (thus the trivial protocol is optimal). This linear lower bound was shown by Nayak [FOCS 1999, pp. 369---376, 1999] to hold also in the quantum setting. Here, we extend Nayak's result by considering approximate privacy, and requiring security only against specious adversaries, which are, in analogy to classical honest-but-curious adversaries, the weakest reasonable quantum adversaries. We show that, even in this weakened scenario, quantum private information retrieval (QPIR) requires $$n$$ n qubits of communication. From this follows that Le Gall's recent QPIR protocol with sublinear communication complexity [Theory Comput. 8(1):369---374, 2012] is not information-theoretically private, against the weakest reasonable cryptographic adversary.

Exponential Separation of Information and Communication for Boolean Functions

Abstract: We show an exponential gap between communication complexity and information complexity for boolean functions, by giving an explicit example of a partial function with information complexity ≤ O(k), and distributional communication complexity ≥ 2k. This shows that a communication protocol for a partial boolean function cannot always be compressed to its internal information. By a result of Braverman [Bra12], our gap is the largest possible. By a result of Braverman and Rao [BR11], our example shows a gap between communication complexity and amortized communication complexity, implying that a tight direct sum result for distributional communication complexity of boolean functions cannot hold, answering a long standing open problem. Our techniques build on [GKR14], that proved a similar result for relations with very long outputs (double exponentially long in k). In addition to the stronger result, the current work gives a simpler proof, benefiting from the short output length of boolean functions.Another (conceptual) contribution of our work is the relative discrepancy method, a new rectangle-based method for proving communication complexity lower bounds for boolean functions, powerful enough to separate information complexity and communication complexity.

Randomized Proof-Labeling Schemes

Abstract: Proof-labeling schemes, introduced by Korman, Kutten and Peleg [PODC 2005], are a mechanism to certify that a network configuration satisfies a given boolean predicate. Such mechanisms find applications in many contexts, e.g., the design of fault-tolerant distributed algorithms. In a proof-labeling scheme, predicate verification consists of neighbors exchanging labels, whose contents depends on the predicate. In this paper, we introduce the notion of randomized proof-labeling schemes where messages are randomized and correctness is probabilistic. We show that randomization reduces label size exponentially while guaranteeing probability of correctness arbitrarily close to one. In addition, we present a novel label-size lower bound technique that applies to both deterministic and randomized proof-labeling schemes. Using this technique, we establish several tight bounds on the verification complexity of MST, acyclicity, connectivity, and longest cycle size.

Interactive Information Complexity

Abstract: The primary goal of this paper is to define and study the interactive information complexity of functions. Let $f(x,y)$ be a function, and suppose Alice is given $x$ and Bob is given $y$. Informally, the interactive information complexity $\mathsf{IC}(f)$ of $f$ is the least amount of information Alice and Bob need to reveal to each other to compute $f$. Previously, information complexity has been defined with respect to a prior distribution on the input pairs $(x,y)$. Our first goal is to give a definition that is independent of the prior distribution. We show that several possible definitions are essentially equivalent. We establish some basic properties of the interactive information complexity $\mathsf{IC}(f)$. In particular, we show that $\mathsf{IC}(f)$ is equal to the amortized (randomized) communication complexity of $f$. We also show a direct sum theorem for $\mathsf{IC}(f)$ and give the first general connection between information complexity and (nonamortized) communication complexity. This conn...

Tighter low-rank approximation via sampling the leveraged element

Abstract: In this work, we propose a new randomized algorithm for computing a low-rank approximation to a given matrix. Taking an approach different from existing literature, our method first involves a specific biased sampling, with an element being chosen based on the leverage scores of its row and column, and then involves weighted alternating minimization over the factored form of the intended low-rank matrix, to minimize error only on these samples. Our method can leverage input sparsity, yet produce approximations in spectral (as opposed to the weaker Frobenius) norm; this combines the best aspects of otherwise disparate current results, but with a dependence on the condition number κ = σ1/σr. In particular we require O(nnz(M)+nκ2r5/e2) computations to generate a rank-r approximation to M in spectral norm. In contrast, the best existing method requires O(nnz(M)+nr2/e4) time to compute an approximation in Frobenius norm. Besides the tightness in spectral norm, we have a better dependence on the error e. Our method is naturally and highly parallelizable.Our new approach enables two extensions that are interesting on their own. The first is a new method to directly compute a low-rank approximation (in efficient factored form) to the product of two given matrices; it computes a small random set of entries of the product, and then executes weighted alternating minimization (as before) on these. The sampling strategy is different because now we cannot access leverage scores of the product matrix (but instead have to work with input matrices). The second extension is an improved algorithm with smaller communication complexity for the distributed PCA setting (where each server has small set of rows of the matrix, and want to compute low rank approximation with small amount of communication with other servers).

An Interactive Information Odometer and Applications

Abstract: We introduce a novel technique which enables two players to maintain an estimate of the internal information cost of their conversation in an online fashion without revealing much extra information. We use this construction to obtain new results about communication complexity and information-theoretic privacy.As a first corollary, we prove a strong direct product theorem for communication complexity in terms of information complexity: If I bits of information are required for solving a single copy of f under μ with probability 2/3, then any protocol attempting to solve n independent copies of f under μn using o(n • I) communication, will succeed with probability 2-Ω(n). This is tight, as Braverman and Rao [BR11] previously showed that O(n • I) communication suffice to succeed with probability ~(2/3)n.We then show how the information odometer can be used to achieve the best possible information-theoretic privacy between two untrusted parties: If the players' goal is to compute a function f(x,y), and f admits a protocol with information cost is I and communication cost C, then our odometer can be used to produce a "robust" protocol which: (i) Assuming both players are honest, computes f with high probability, and (ii) Even if one party is malicious, then for any k∈N, the probability that the honest player reveals more than O(k • (I+log C)) bits of information to the other player is at most 2-Ω(k).Finally, we outline an approach which uses the odometer as a proxy for breaking state of the art interactive compression results: We show that our odometer allows to reduce interactive compression to the regime where I=O(log C), thereby opening a potential avenue for improving the compression result of [BBCR10] and to new direct sum and product theorems in communication complexity.

Lower Bounds for Clique vs. Independent Set.

Abstract: We prove a super logarithmic lower bound on the Conon deterministic communication complexity of the Clique vs. Independent Set problem introduced by Yannakakis (STOC 1988, JCSS 1991). As a corollary, this implies super polynomial lower bounds for the Alon -- Saks -- Seymour conjecture in graph theory. Our approach is to first exhibit a query complexity separation for the decision tree analogue of the UP vs. coNP question -- namely, unambiguous DNF width vs. CNF width -- and then "lift" this separation over to communication complexity using a result from prior work.

Efficient Asynchronous Verifiable Secret Sharing and Multiparty Computation

Abstract: Secure Multi-Party Computation (MPC) providing information-theoretic security allows a set of n parties to securely compute an agreed function over a finite field, even if t parties are under the control of a computationally unbounded active adversary. Asynchronous MPC (AMPC) is an important variant of MPC, which works over an asynchronous network. It is well known that perfect AMPC is possible if and only if t

Communication Complexity of Distributed Convex Learning and Optimization

Abstract: We study the fundamental limits to communication-efficient distributed methods for convex learning and optimization, under different assumptions on the information available to individual machines, and the types of functions considered. We identify cases where existing algorithms are already worst-case optimal, as well as cases where room for further improvement is still possible. Among other things, our results indicate that without similarity between the local objective functions (due to statistical data similarity or otherwise) many communication rounds may be required, even if the machines have unbounded computational power.