Algorithms and Lower Bounds for Comparator Circuits from Shrinkage

Abstract

AbstractIn this paper, we initiate the study of average-case complexity and circuit analysis algorithms for comparator circuits. Departing from previous approaches, we exploit the technique of shrinkage under random restrictions to obtain a variety of new results for this model. Among them, we show Average-case Lower Bounds For every $$k = k(n)$$ k = k ( n ) with $$k \geqslant \log n$$ k ⩾ log n , there exists a polynomial-time computable function $$f_k$$ f k on n bits such that, for every comparator circuit C with at most $$n^{1.5}/O\!\left( k\cdot \sqrt{\log n}\right) $$ n 1.5 / O k · log n gates, we have $$\begin{aligned} \mathop {{{\,\mathrm{\textbf{Pr}}\,}}}\limits _{x\in \left\{ 0,1\right\} ^n}\left[ C(x)=f_k(x)\right] \leqslant \frac{1}{2} + \frac{1}{2^{\Omega (k)}}. \end{aligned}$$ Pr x ∈ 0 , 1 n C ( x ) = f k ( x ) ⩽ 1 2 + 1 2 Ω ( k ) . This average-case lower bound matches the worst-case lower bound of Gál and Robere by letting $$k=O\!\left( \log n\right) $$ k = O log n . $$\#$$ # SAT Algorithms There is an algorithm that counts the number of satisfying assignments of a given comparator circuit with at most $$n^{1.5}/O\!\left( k\cdot \sqrt{\log n}\right) $$ n 1.5 / O k · log n gates, in time $$2^{n-k}\cdot {{\,\textrm{poly}\,}}(n)$$ 2 n - k · poly ( n ) , for any $$k\leqslant n/4$$ k ⩽ n / 4 . The running time is non-trivial (i.e., $$2^n/n^{\omega (1)}$$ 2 n / n ω ( 1 ) ) when $$k=\omega (\log n)$$ k = ω ( log n ) . Pseudorandom Generators and $$\textsf {MCSP} $$ MCSP Lower Bounds There is a pseudorandom generator of seed length $$s^{2/3+o(1)}$$ s 2 / 3 + o ( 1 ) that fools comparator circuits with s gates. Also, using this PRG, we obtain an $$n^{1.5-o(1)}$$ n 1.5 - o ( 1 ) lower bound for $$\textsf {MCSP} $$ MCSP against comparator circuits.