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

Spectral Norm of Symmetric Functions

The spectral norm of a Boolean function $f:\{0,1\}^n \to \{-1,1\}$ is the sum of the absolute values of its Fourier coefficients. This quantity provides useful upper and lower bounds on the complexity of a function in areas such as learning theory, circuit complexity, and communication complexity. In this paper, we give a combinatorial characterization for the spectral norm of symmetric functions. We show that the logarithm of the spectral norm is of the same order of magnitude as $r(f)\log(n/r(f))$ where $r(f) = \max\{r_0,r_1\}$, and $r_0$ and $r_1$ are the smallest integers less than $n/2$ such that $f(x)$ or $f(x) \cdot parity(x)$ is constant for all $x$ with $\sum x_i \in [r_0, n-r_1]$. We mention some applications to the decision tree and communication complexity of symmetric functions

Some results on circuit depth

An important problem in theoretical computer science is to develop methods for estimating the complexity of finite functions. For many familiar functions there remain important gaps between the best known lower and upper bound we investigate the inherent complexity of Boolean functional taking circuits as our model of computation and depth (or delay)to be the measure of complexity. The relevance of circuits as a model of computation for Boolean functions stems from the fact that Turing machine computations may be efficiently simulated by circuits. Important relations among various measures of circuit complexity are btained as well as bounds on the maximum depth of any function and of any monotone function. We then give a detailed account of the complexity of NAND circuits for several important functions and pursue an analysis of the important set of symmetric functions. A number of gap theorems for symmetric functions are exhibited and these are contrasted with uniform hierarchies for several large sets of functions. Finally, we describe several short formulae for threshold functions

Barriers for Rank Methods in Arithmetic Complexity

Arithmetic complexity, the study of the cost of computing polynomials via additions and multiplications, is considered (for many good reasons) simpler to understand than Boolean complexity, namely computing Boolean functions via logical gates. And indeed, we seem to have significantly more lower bound techniques and results in arithmetic complexity than in Boolean complexity. Despite many successes and rapid progress, however, foundational challenges, like proving super-polynomial lower bounds on circuit or formula size for explicit polynomials, or super-linear lower bounds on explicit 3-dimensional tensors, remain elusive. At the same time (and possibly for similar reasons), we have plenty more excuses, in the form of "barrier results" for failing to prove basic lower bounds in Boolean complexity than in arithmetic complexity. Efforts to find barriers to arithmetic lower bound techniques seem harder, and despite some attempts we have no excuses of similar quality for these failures in arithmetic complexity. This paper aims to add to this study. In this paper we address rank methods, which were long recognized as encompassing and abstracting almost all known arithmetic lower bounds to-date, including the most recent impressive successes. Rank methods (under the name of flattenings) are also in wide use in algebraic geometry for proving tensor rank and symmetric tensor rank lower bounds. Our main results are barriers to these methods. In particular, 1. Rank methods cannot prove better than (2^d)*n^(d/2) lower bound on the tensor rank of any d-dimensional tensor of side n. (In particular, they cannot prove super-linear, indeed even >8n tensor rank lower bounds for any 3-dimensional tensors.) 2. Rank methods cannot prove (d+1)n^(d/2) on the Waring rank of any n-variate polynomial of degree d. (In particular, they cannot prove such lower bounds on stronger models, including depth-3 circuits.) The proofs of these bounds use simple linear-algebraic arguments, leveraging connections between the symbolic rank of matrix polynomials and the usual rank of their evaluations. These techniques can perhaps be extended to barriers for other arithmetic models on which progress has halted. To see how these barrier results directly inform the state-of-art in arithmetic complexity we note the following. First, the bounds above nearly match the best explicit bounds we know for these models, hence offer an explanations why the rank methods got stuck there. Second, the bounds above are a far cry (quadratically away) from the true complexity (e.g. of random polynomials) in these models, which if achieved (by any methods), are known to imply super-polynomial formula lower bounds. We also explain the relation of our barrier results to other attempts, and in particular how they significantly differ from the recent attempts to find analogues of "natural proofs" for arithmetic complexity. Finally, we discuss the few arithmetic lower bound approaches which fall outside rank methods, and some natural directions our barriers suggest

The Complexity of Symmetry

One of the main goals of theoretical computer science is to prove limits on how efficiently certain Boolean functions can be computed. The study of the algebraic complexity of polynomials provides an indirect approach to exploring these questions, which may prove fruitful since much is known about polynomials already from the field of algebra. This paper explores current research in establishing lower bounds on invariant rings and polynomial families. It explains the construction of an invariant ring for whom a succinct encoding would imply that NP is in P/poly. It then states a theorem about the circuit complexity partial derivatives, and its implications for elementary symmetric function complexity, and proposes potential implications for other classes of functions

The Orthogonal Vectors Conjecture for Branching Programs and Formulas

In the Orthogonal Vectors (OV) problem, we wish to determine if there is an orthogonal pair of vectors among n Boolean vectors in d dimensions. The OV Conjecture (OVC) posits that OV requires n^{2-o(1)} time to solve, for all d=omega(log n). Assuming the OVC, optimal time lower bounds have been proved for many prominent problems in P, such as Edit Distance, Frechet Distance, Longest Common Subsequence, and approximating the diameter of a graph. We prove that OVC is true in several computational models of interest: - For all sufficiently large n and d, OV for n vectors in {0,1}^d has branching program complexity Theta~(n * min(n,2^d)). In particular, the lower and upper bounds match up to polylog factors. - OV has Boolean formula complexity Theta~(n * min(n,2^d)), over all complete bases of O(1) fan-in. - OV requires Theta~(n * min(n,2^d)) wires, in formulas comprised of gates computing arbitrary symmetric functions of unbounded fan-in. Our lower bounds basically match the best known (quadratic) lower bounds for any explicit function in those models. Analogous lower bounds hold for many related problems shown to be hard under OVC, such as Batch Partial Match, Batch Subset Queries, and Batch Hamming Nearest Neighbors, all of which have very succinct reductions to OV. The proofs use a certain kind of input restriction that is different from typical random restrictions where variables are assigned independently. We give a sense in which independent random restrictions cannot be used to show hardness, in that OVC is false in the "average case" even for AC^0 formulas: For all p in (0,1) there is a delta_p > 0 such that for every n and d, OV instances with input bits independently set to 1 with probability p (and 0 otherwise) can be solved with AC^0 formulas of O(n^{2-delta_p}) size, on all but a o_n(1) fraction of instances. Moreover, lim_{p - > 1}delta_p = 1