On ϵ-sensitive monotone computations - Hrubeš, Pavel (Computational complexity, 2020)
We show that strong-enough lower bounds on monotone arithmetic circuits or the nonnegative rank of a matrix imply unconditional lower bounds in arithmetic or Boolean circuit complexity. First, we show that if a polynomial f ∈ R [ x 1 , ⋯ , x n ] of degree d has an arithmetic circuit of size s then ( x 1 + ⋯ + x n + 1 ) d + ϵ f has a monotone arithmetic circuit of size O ( s d 2 + n log n ) , for some ϵ > 0 . Second, if f : { 0 , 1 } n → { 0 , 1 } is a Boolean function, we associate with f an explicit exponential-size matrix M ( f ) such that the Boolean circuit size of f is at least Ω ( min ϵ > 0 ( rk + ( M ( f ) - ϵ J ) ) - 2 n ) , where J is the all-ones matrix and rk + denotes the nonnegative rank of a matrix. In fact, the quantity min ϵ > 0 ( rk + ( M ( f ) - ϵ J ) ) characterizes how hard is it to distinguish rejecting and accepting inputs of f by means of a linear program. Finally, we introduce a proof system resembling the monotone calculus of Atserias et al. (J Comput Syst Sci 65:626–638, 2002) and show that similar ϵ -sensitive lower bounds on monotone arithmetic circuits imply lower bounds on proof-size in the system.