The Landscape of Communication Complexity Classes
Computational Complexity(2018)
摘要
We prove several results which, together with prior work, provide a nearly-complete picture of the relationships among classical communication complexity classes between 𝖯 and 𝖯𝖲𝖯𝖠𝖢𝖤 , short of proving lower bounds against classes for which no explicit lower bounds were already known. Our article also serves as an up-to-date survey on the state of structural communication complexity. Among our new results we show that 𝖬𝖠⊈𝖹𝖯𝖯^𝖭𝖯[1] , that is, Merlin–Arthur proof systems cannot be simulated by zero-sided error randomized protocols with one 𝖭𝖯 query. Here the class 𝖹𝖯𝖯^𝖭𝖯[1] has the property that generalizing it in the slightest ways would make it contain 𝖠𝖬∩𝖼𝗈𝖠𝖬 , for which it is notoriously open to prove any explicit lower bounds. We also prove that 𝖴𝖲⊈𝖹𝖯𝖯^𝖭𝖯[1] , where 𝖴𝖲 is the class whose canonically complete problem is the variant of set-disjointness where yes-instances are uniquely intersecting. We also prove that 𝖴𝖲⊈𝖼𝗈𝖣𝖯 , where 𝖣𝖯 is the class of differences of two 𝖭𝖯 sets. Finally, we explore an intriguing open issue: Are rank-1 matrices inherently more powerful than rectangles in communication complexity? We prove a new separation concerning 𝖯𝖯 that sheds light on this issue and strengthens some previously known separations.
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关键词
landscape,communication,complexity,classes
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