Some Results in Circuit Complexity

The thesis consists of three parts. The first part investigates the AC complexity of subgraph isomorphism problem that is, detecting whether an n-vertex input graph contains a P -subgraph for some fixed P . For the average-case complexity, where the input is a distribution on random graphs at the threshold and small error is allowed, we are able to explicitly characterize the minimal AC size up to a quadratic factor. For the worst-case complexity, if P is a colored graph (each vertex has a different color, and the input graph is also colored, of course), we prove a n )/ log tw(P )) lower bound, which nearly matches the upper bound n )+O(1) due to Alon, Yuster and Zwick [5]. For the uncolored worst-case complexity, we prove that there is no monotone projection whatsoever that reduces Subgraph(M3) to Subgraph(P3 . + M2) (P3 is a path on 3 vertices, Mk is a matching with k edges, and “ . +” stands for the disjoint union), which suggests that the complexity of the uncolored case is not minor monotone (unlike the colored case). The second part proves tight lower bounds on the immunity of MOD function and its negation. For a Boolean function f : {0, 1} → {0, 1}, its immunity over the field Fp is defined as the minimal degree of a nontrivial polynomial g(x) ∈ Fp[x1, . . . , xn] such that g(x) = 0 for all x with f(x) = 0. Improving the previous results, we prove: when p, q are coprime, the immunity of ¬MODq is exactly ⌊(n + q − 1)/q⌋; and the immunity of MODq is lower bounded by ⌈n/2⌉. We observe the following connection between immunity and circuit lower bounds: if Boolean function f has immunity over Fp at least n/2− o( √ n) and |1f | = Ω(2), then f requires exponential size AC[p] (=AC with MODp gates) to compute. The third part is a complete characterization of k robust immune symmetric Boolean functions for any k, over any field, where a Boolean function f : {0, 1} → {0, 1} is k robust immune if the immunity of f and 1 − f is always lower bounded by k no matter how you change the values of f(x) with k ≤ |x| ≤ n− k. The first part is joint with Alexander Razborov and Benjamin Rossman, and the second part is joint with Chris Beck.