Communication complexity

About: Communication complexity is a research topic. Over the lifetime, 3870 publications have been published within this topic receiving 105832 citations.

On the complexity of price equilibria

Abstract: We prove complexity, approximability, and inapproximability results for the problem of finding an exchange equilibrium in markets with indivisible (integer) goods, most notably a polynomial algorithm that approximates the market equilibrium arbitrarily close when the number of goods is bounded and the utilities linear. We also show a communication complexity lower bound in a model appropriate for markets. Our result implies that the ideal informational economy of a market with divisible goods and unique optimal allocations is unattainable in general.

On the virtue of succinct proofs: amplifying communication complexity hardness to time-space trade-offs in proof complexity

Abstract: An active line of research in proof complexity over the last decade has been the study of proof space and trade-offs between size and space. Such questions were originally motivated by practical SAT solving, but have also led to the development of new theoretical concepts in proof complexity of intrinsic interest and to results establishing nontrivial relations between space and other proof complexity measures. By now, the resolution proof system is fairly well understood in this regard, as witnessed by a sequence of papers leading up to [Ben-Sasson and Nordstrom 2008, 2011] and [Beame, Beck, and Impagliazzo 2012]. However, for other relevant proof systems in the context of SAT solving, such as polynomial calculus (PC) and cutting planes (CP), very little has been known. Inspired by [BN08, BN11], we consider CNF encodings of so-called pebble games played on graphs and the approach of making such pebbling formulas harder by simple syntactic modifications. We use this paradigm of hardness amplification to make progress on the relatively longstanding open question of proving time-space trade-offs for PC and CP. Namely, we exhibit a family of modified pebbling formulas {F_n} such that: - The formulas F_n have size O(n) and width O(1). - They have proofs in length O(n) in resolution, which generalize to both PC and CP. - Any refutation in CP or PCR (a generalization of PC) in length L and space s must satisfy s log L >≈ √[4]{n}. A crucial technical ingredient in these results is a new two-player communication complexity lower bound for composed search problems in terms of block sensitivity, a contribution that we believe to be of independent interest.

Super-logarithmic Depth Lower Bounds via Direct Sum in Communication Complexity (Preliminary Version)

Abstract: Is it easier to solve two communication problems to- gether than separately? This question is related to the complexity of the composition of boolean functions. Based on this relationship, an approach to separating NC' from P is outlined. Furthermore, it is shown that the approach provides a new proof of the separation of monotone NC' from monotone P.

Sense of direction, topological awareness and communication complexity

Abstract: Based on some recent results, it is here argued that the communication complexity of distributed problems can be greatly affected by two factors hereby identified as 'sense of direction' and 'topological awareness'. It is also suggested that 'insensitivity' to either or both factors is an indicator of the inherent difficulty of a distributed problem. A bibliography of recent results is included.

Hellinger Strikes Back: A Note on the Multi-party Information Complexity of AND

Abstract: The ${\textsc{And}}$ problem on t bits is a promise decision problem where either at most one bit of the input is set to 1 ( No instance) or all t bits are set to 1 (${\textsc{Yes}}$ instance) In this note, I will give a new proof of an ***(1/t ) lower bound on the information complexity of ${\textsc{And}}$ in the number-in-hand model of communication This was recently established by Gronemeier, STACS 2009 The proof exploits the information geometry of communication protocols via Hellinger distance in a novel manner and avoids the analytic approach inherent in previous work As previously known, this bound implies an ***(n /t ) lower bound on the communication complexity of multiparty disjointness and consequently a ***(n 1 *** 2/k ) space lower bound on estimating the k -th frequency moment F k