Algorithms and Lower Bounds for Comparator Circuits from Shrinkage

Abstract

In 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 showAverage-case Lower Bounds For every k = k(n) with k geqslant log n, there exists a polynomial-time computable function f_k on n bits such that, for every comparator circuit C with at most n^{1.5}/O!left( kcdot sqrt{log n}right) gates, we have Prx∈0,1nC(x)=fk(x)⩽12+12Ω(k).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\mathop {{{\\,\\mathrm{\ extbf{Pr}}\\,}}}\\limits _{x\\in \\left\\{ 0,1\\right\\} ^n}\\left[ C(x)=f_k(x)\\right] \\leqslant \\frac{1}{2} + \\frac{1}{2^{\\Omega (k)}}. \\end{aligned}$$\\end{document} This average-case lower bound matches the worst-case lower bound of Gál and Robere by letting k=O!left( log nright) .#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( kcdot sqrt{log n}right) gates, in time 2^{n-k}cdot {{,textrm{poly},}}(n), for any kleqslant n/4. The running time is non-trivial (i.e., 2^n/n^{omega (1)}) when k=omega (log n).Pseudorandom Generators and textsf {MCSP} Lower Bounds There is a pseudorandom generator of seed length s^{2/3+o(1)} that fools comparator circuits with s gates. Also, using this PRG, we obtain an n^{1.5-o(1)} lower bound for textsf {MCSP} against comparator circuits.