Algorithms and Lower Bounds in Circuit Complexity

Computational complexity theory aims to understand what problems can be efficiently solved by computation. This thesis studies computational complexity in the model of Boolean circuits. Boolean circuits provide a basic mathematical model for computation and play a central role in complexity theory, with important applications in separations of complexity classes, algorithm design, and pseudorandom constructions. In this thesis, we investigate various types of circuit models such as threshold circuits, Boolean formulas, and their extensions, focusing on obtaining complexity-theoretic lower bounds and algorithmic upper bounds for these circuits. (1) Algorithms and lower bounds for generalized threshold circuits: We extend the study of linear threshold circuits, circuits with gates computing linear threshold functions, to the more powerful model of polynomial threshold circuits where the gates can compute polynomial threshold functions. We obtain hardness and meta-algorithmic results for this circuit model, including strong average-case lower bounds, satisfiability algorithms, and derandomization algorithms for constant-depth polynomial threshold circuits with super-linear wire complexity. (2) Algorithms and lower bounds for enhanced formulas: We investigate the model of Boolean formulas whose leaf gates can compute complex functions. In particular, we study De Morgan formulas whose leaf gates are functions with "low communication complexity". Such gates can capture a broad class of functions including symmetric functions and polynomial threshold functions. We obtain new and improved results in terms of lower bounds and meta-algorithms (satisfiability, derandomization, and learning) for such enhanced formulas. (3) Circuit lower bounds for MCSP: We study circuit lower bounds for the Minimum Circuit Size Problem (MCSP), the fundamental problem of deciding whether a given function (in the form of a truth table) can be computed by small circuits. We get new and improved lower bounds for MCSP that nearly match the best-known lower bounds against several well-studied circuit models such as Boolean formulas and constant-depth circuits

Algorithms and lower bounds for de Morgan formulas of low-communication leaf gates

The class $FORMULA[s] \circ \mathcal{G}$ consists of Boolean functions computable by size-$s$ de Morgan formulas whose leaves are any Boolean functions from a class $\mathcal{G}$. We give lower bounds and (SAT, Learning, and PRG) algorithms for $FORMULA[n^{1.99}]\circ \mathcal{G}$, for classes $\mathcal{G}$ of functions with low communication complexity. Let $R^{(k)}(\mathcal{G})$ be the maximum $k$-party NOF randomized communication complexity of $\mathcal{G}$. We show: (1) The Generalized Inner Product function $GIP^k_n$ cannot be computed in $FORMULA[s]\circ \mathcal{G}$ on more than $1/2+\varepsilon$ fraction of inputs for $$s = o \! \left ( \frac{n^2}{ \left(k \cdot 4^k \cdot {R}^{(k)}(\mathcal{G}) \cdot \log (n/\varepsilon) \cdot \log(1/\varepsilon) \right)^{2}} \right).$$ As a corollary, we get an average-case lower bound for $GIP^k_n$ against $FORMULA[n^{1.99}]\circ PTF^{k-1}$. (2) There is a PRG of seed length $n/2 + O\left(\sqrt{s} \cdot R^{(2)}(\mathcal{G}) \cdot\log(s/\varepsilon) \cdot \log (1/\varepsilon) \right)$ that $\varepsilon$-fools $FORMULA[s] \circ \mathcal{G}$. For $FORMULA[s] \circ LTF$, we get the better seed length $O\left(n^{1/2}\cdot s^{1/4}\cdot \log(n)\cdot \log(n/\varepsilon)\right)$. This gives the first non-trivial PRG (with seed length $o(n)$) for intersections of $n$ half-spaces in the regime where $\varepsilon \leq 1/n$. (3) There is a randomized $2^{n-t}$-time $\#$SAT algorithm for $FORMULA[s] \circ \mathcal{G}$, where $$t=\Omega\left(\frac{n}{\sqrt{s}\cdot\log^2(s)\cdot R^{(2)}(\mathcal{G})}\right)^{1/2}.$$ In particular, this implies a nontrivial #SAT algorithm for $FORMULA[n^{1.99}]\circ LTF$. (4) The Minimum Circuit Size Problem is not in $FORMULA[n^{1.99}]\circ XOR$. On the algorithmic side, we show that $FORMULA[n^{1.99}] \circ XOR$ can be PAC-learned in time $2^{O(n/\log n)}$