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Christopher Rosin

Publications and source records attributed to Christopher Rosin.

2 recordsLinked to original sources
arXiv2026

Optimal Monotone Depth-Three Circuit Lower Bounds for Majority

Gurumuhkani et al. (CCC'24) introduced the local enumeration problem $Enum(k, t)$ as follows: for a natural number $k$ and a parameter $t$, given an $n$-variate $k$-CNF with no satisfying assignment with Hamming weight less than $t(n)$, enumerate all satisfying assignments of Hamming weight exactly $t(n)$. They showed that efficient algorithms for local enumeration yield new $k$-SAT algorithms and depth-3 lower bounds for Majority function. As the first non-trivial case, they gave an algorithm for $k = 3$ which in particular gave a new lower bound on the size of depth-3 circuits with bottom fan-in at most 3 computing Majority. In this paper, we give an optimal algorithm that solves local enumeration on monotone formulas for $k = 3$ and all $t \le n/2$. In particular, we obtain an optimal lower bound on the size of monotone depth-3 circuits with bottom fan-in at most 3 computing Majority.

cs.CC↗
arXiv2026

Optimal Depth-Three Circuits for Inner Product

We show that Inner Product in $2n$ variables, $\mathbf{IP}_n(x, y) = x_1y_1 \oplus \ldots \oplus x_ny_n$, can be computed by depth-3 bottom fan-in 2 circuits of size $\mathsf{poly}(n)\cdot (9/5)^n$, matching the lower bound of G\"o\"os, Guan, and Mosnoi (Inform. Comput.'24). Our construction is obtained via the following steps. - We provide a general template for constructing optimal depth-3 circuits with bottom fan-in $k$ for an arbitrary function $f$. We do this in two steps. First, we partition $f^{-1}(1)$ into orbits of its automorphism group. Second, for each orbit, we construct one $k$-CNF that (a) accepts the largest number of inputs from that orbit and (b) rejects all inputs rejected by $f$. - We instantiate the template for $\mathbf{IP}_n$ and $k = 2$. Guided by the intuition (which we call modularity principle) that optimal 2-CNFs can be constructed by taking the conjunction of variable-disjoint copies of smaller $2$-CNFs, we use computer search to identify a small set of building block 2-CNFs over at most 4 variables. - We again use computer search to discover appropriate combinations (disjoint conjunctions) of building blocks to arrive at optimal 2-CNFs and analyze them using techniques from analytic combinatorics. We believe that the approach outlined in this paper can be applied to a wide range of functions to determine their depth-3 complexity.

cs.CC↗