Near-optimal small-depth lower bounds for small distance connectivity

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 nΩ(k1/d/d) when k(n) ≤ n1/5, and nΩ(k1/5d/d) when k(n)≤ n. The previous best circuit size lower bounds were nkexp(−O(d)) (by Beame, Impagliazzo, and Pitassi (Computational Complexity 1998)) and nΩ((logk)/d) (following from a recent formula size lower bound of Rossman (STOC 2014)). Our lower bound is quite close to optimal, as a simple construction gives depth-d circuits of size nO(k2/d) for this problem (and strengthening our bound even to nkΩ(1/d) would require proving that undirected connectivity is not in NC1). Our proof is by reduction to a new lower bound on the size of small-depth circuits computing a skewed variant of the “Sipser functions” that have played an important role in classical circuit lower bounds. A key ingredient in our proof of the required lower bound for these Sipser-like functions is the use of random projections, an extension of random restrictions which were recently employed by Rossman, Servedio, and Tan (FOCS 2015). Random projections allow us to obtain sharper quantitative bounds while employing simpler arguments, both conceptually and technically, than in the previous works.