
A Solution of the P versus NP Problem based on specific property of clique function
Circuit lower bounds are important since it is believed that a superpol...
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A Solution of the P versus NP Problem
Berg and Ulfberg and Amano and Maruoka have used CNFDNFapproximators t...
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Monotone Circuit Lower Bounds from Robust Sunflowers
Robust sunflowers are a generalization of combinatorial sunflowers that ...
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From expanders to hitting distributions and simulation theorems
Recently, Chattopadhyay et al. [chattopadhyay 2017 simulation] proved th...
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Stochastically Controlled Stochastic Gradient for the Convex and Nonconvex Composition problem
In this paper, we consider the convex and nonconvex composition problem...
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From DNF compression to sunflower theorems via regularity
The sunflower conjecture is one of the most wellknown open problems in ...
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Higher order monotonicity and submodularity of influence in social networks: from local to global
Kempe, Kleinberg and Tardos (KKT) proposed the following conjecture abou...
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KRW Composition Theorems via Lifting
One of the major open problems in complexity theory is proving superlogarithmic lower bounds on the depth of circuits (i.e., πβππ^1). Karchmer, Raz, and Wigderson (Computational Complexity 5(3/4), 1995) suggested to approach this problem by proving that depth complexity behaves "as expected" with respect to the composition of functions fβ’ g. They showed that the validity of this conjecture would imply that πβππ^1. Several works have made progress toward resolving this conjecture by proving special cases. In particular, these works proved the KRW conjecture for every outer function f, but only for few inner functions g. Thus, it is an important challenge to prove the KRW conjecture for a wider range of inner functions. In this work, we extend significantly the range of inner functions that can be handled. First, we consider the monotone version of the KRW conjecture. We prove it for every monotone inner function g whose depth complexity can be lower bounded via a querytocommunication lifting theorem. This allows us to handle several new and wellstudied functions such as the stconnectivity, clique, and generation functions. In order to carry this progress back to the nonmonotone setting, we introduce a new notion of semimonotone composition, which combines the nonmonotone complexity of the outer function f with the monotone complexity of the inner function g. In this setting, we prove the KRW conjecture for a similar selection of inner functions g, but only for a specific choice of the outer function f.
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