# Communication Complexity and Applications (14w5164)

Arriving in Banff, Alberta Sunday, August 24 and departing Friday August 29, 2014

## Organizers

(Dartmouth College)

(Simon Fraser University)

(University of Massachusetts Amherst)

(University of Washington)

## Objectives

Communication complexity has proven to be a very useful abstraction for studying computational processes and the interactions of their internal components, as well as interactions between separate processes. It is hard to imagine a computational process that does not involve some form of communication. For example, one can take any computational device (such as a cellphone or a computer), divide it into two parts, and then view the device as executing a communication protocol between these parts. In this way, a lower bound on the communication required can be translated into a lower bound on the resources (for example, the number of wires) needed to build the device. Likewise, any distributed architecture (be it a multi-core processor or the entire Internet) naturally represents the execution of a communication protocol amongst the constituent processes.

Over the past three decades, the communication complexity model has been applied to the study of a wide array of subfields of computation. Recently, the model has been exploited to spectacular effect in the study of space-efficient data stream algorithms. Given a streaming problem, one can view it as being solved by communicating processes as follows. One can partition the stream into two (or more) parts, and view the contents of the memory during the execution of the algorithm as a communication between the parts; i.e., the contents of the memory at a given time are the same as the contents of a message being passed between two processes. Indeed, lower bounds on the space requirements of data-stream algorithms are almost exclusively proved by appealing to communication bounds.

Another application for communication complexity techniques is in property testing. Recent results have established a close link between communication complexity and property testing. Here we consider multiple parties who communicate to compute a shared function which is closely linked to the property being tested. The communication between the parties corresponds to the queries being made by a tester for the corresponding property; thus, lower bounds for testing the property are implied by the communication required by the communicating parties.

The last twenty years have witnessed a flurry of research activity around communication complexity. A wide range of techniques for proving communication lower bounds have been established. Examples include linear algebraic methods based on factorization norms, singular value estimation, and semidefinite programming, combinatorial methods based on monochromatic and near-monochromatic rectangles, and information-theoretic methods based on entropy and mutual information. Further, the subject has branched out into quantum communication complexity, which has provided its own set of unique insights and ideas. Significant progress has been made towards unifying these approaches via the information-theoretic approach. In addition to being an important step in establishing a unifying theory, these advances have brought us closer to solving longstanding open questions in communication complexity, such as the direct-sum and direct-product questions.

In the direct-sum problem, we consider the complexity of computing n copies of a function, in terms of the complexity of a single copy. For example, known fast matrix multiplication algorithms demonstrate an "economy of scale" phenomenon: there exists a linear function L on n bits for which the smallest circuit computing L(x) has size at least $n^2$, whereas the smallest circuit computing $L(x1)$, $L(x2)$, ..., $L(xn)$ for n different inputs is of size $n^2.4$. A central open question is whether a similar phenomenon exists in communication complexity. Advances in recent years have used information-theoretic methods to give interesting answers to this fundamental question. In particular, a near-equivalent formulation is whether it is possible to compress interactive communication to a size proportional to the information content of the communication. If so, such a result would generalize classical results of Shannon and Huffman in the non-interactive setting.

Concurrent with research in communication complexity, the study of data stream algorithms has achieved significant maturity in the sense that we now have a comprehensive understanding of estimating "frequency-based" statistics such as quantiles and other functions of the distribution of elements in the stream. Space-efficient algorithms have been designed that have been shown to be optimal by appealing to communication complexity lower bounds. However, the focus in stream processing has now moved to significantly more structured problems such as linear-algebraic problems (e.g., low-rank approximation, regression) and graph problems (e.g., spectral sparsification, analyzing matchings and related polytopes, spanners and distances). Understanding the complexity of problems that do not cleanly decompose into "primitive problems" (as was the case in frequency-based statistics) is an important direction for future research. In addition, new applications have been discovered for communication complexity: for example, we have the aforementioned application to property testing, and also applications to the design of data structures and compressed sensing.

Although the theory of communication complexity is now closely connected with applications to space-efficient algorithms, the research communities in the respective areas have remained segregated, being naturally sub-communities within "complexity theory" and "algorithms" respectively. An important goal of this workshop is to remedy this situation and foster closer ties. This also involves training young researchers who will become skilled in both areas.

More technical goals of this workshop include the following. 1) Identifying promising approaches to resolving core open problems in communication complexity such as the direct-sum question and the log-rank conjecture. 2) Extending the clean setting of direct-sum to cover general combiner functions including partial functions, as well as primitive problems that can not be made fully independent. Collaborations with applications researchers would help identify the most interesting extensions to study. 3) Identifying new communication problems arising in well-established application areas such as data streams and data structures. 4) Strengthening the newly formed connections of communication complexity to the field of property testing. 5) Furthering the study of Arthur-Merlin communication complexity, with a view towards applications to data stream algorithms that can interact with a powerful helper.