An Exponential Lower Bound on the Size of Constant-Depth Threshold Circuits with Small Energy Complexity

2007 ◽  
Author(s):  
Kei Uchizawa ◽  
Eiji Takimoto
Keyword(s):  
Lower Bound ◽  
Constant Depth ◽  
Small Energy ◽  
Threshold Circuits ◽  
Exponential Lower Bound

Lower bounds for cutting planes proofs with small coefficients

1997 ◽  
Vol 62 (3) ◽  
pp. 708-728 ◽  
Author(s):  
Maria Bonet ◽  
Toniann Pitassi ◽  
Ran Raz
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Cutting Planes ◽  
Communication Complexity ◽  
Proof System ◽  
Small Weight ◽  
Deduction Rule ◽  
Special Case ◽  
Exponential Lower Bound

AbstractWe consider small-weight Cutting Planes (CP*) proofs; that is, Cutting Planes (CP) proofs with coefficients up to Poly(n). We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP* proofs, for a family of tautologies based on the clique function. Because Resolution is a special case of small-weight CP, our method also gives a new and simpler exponential lower bound for Resolution.We also prove the following two theorems: (1) Tree-like CP* proofs cannot polynomially simulate non-tree-like CP* proofs. (2) Tree-like CP* proofs and Bounded-depth-Frege proofs cannot polynomially simulate each other.Our proofs also work for some generalizations of the CP* proof system. In particular, they work for CP* with a deduction rule, and also for any proof system that allows any formula with small communication complexity, and any set of sound rules of inference.

scholarly journals An Exponential Lower Bound for Cut Sparsifiers in Planar Graphs

Algorithmica ◽  
2018 ◽  
Vol 81 (10) ◽  
pp. 4029-4042 ◽  
Author(s):  
Nikolai Karpov ◽  
Marcin Pilipczuk ◽  
Anna Zych-Pawlewicz
Keyword(s):  
Lower Bound ◽  
Planar Graphs ◽  
Exponential Lower Bound
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