Nearoptimal smalldepth lower bounds for small distance connectivity
Abstract
We show that any depth$d$ circuit for determining whether an $n$node graph has an $s$to$t$ path of length at most $k$ must have size $n^{\Omega(k^{1/d}/d)}$. The previous best circuit size lower bounds for this problem were $n^{k^{\exp(O(d))}}$ (due to Beame, Impagliazzo, and Pitassi [BIP98]) and $n^{\Omega((\log k)/d)}$ (following from a recent formula size lower bound of Rossman [Ros14]). Our lower bound is quite close to optimal, since a simple construction gives depth$d$ circuits of size $n^{O(k^{2/d})}$ for this problem (and strengthening our bound even to $n^{k^{\Omega(1/d)}}$ would require proving that undirected connectivity is not in $\mathsf{NC^1}.$) Our proof is by reduction to a new lower bound on the size of smalldepth circuits computing a skewed variant of the "Sipser functions" that have played an important role in classical circuit lower bounds [Sip83, Yao85, Hås86]. A key ingredient in our proof of the required lower bound for these Sipserlike functions is the use of \emph{random projections}, an extension of random restrictions which were recently employed in [RST15]. Random projections allow us to obtain sharper quantitative bounds while employing simpler arguments, both conceptually and technically, than in the previous works [Ajt89, BPU92, BIP98, Ros14].
 Publication:

arXiv eprints
 Pub Date:
 September 2015
 arXiv:
 arXiv:1509.07476
 Bibcode:
 2015arXiv150907476C
 Keywords:

 Computer Science  Computational Complexity