1,068 research outputs found
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
Average-Case Lower Bounds and Satisfiability Algorithms for Small Threshold Circuits
We show average-case lower bounds for explicit Boolean functions against bounded-depth threshold circuits with a superlinear number of wires. We show that for each integer d > 1, there is epsilon_d > 0 such that Parity has correlation at most 1/n^{Omega(1)} with depth-d threshold circuits which have at most n^{1+epsilon_d} wires, and the Generalized Andreev Function has correlation at most 1/2^{n^{Omega(1)}} with depth-d threshold circuits which have at most n^{1+epsilon_d} wires. Previously, only worst-case lower bounds in this setting were known [Impagliazzo/Paturi/Saks, SIAM J. Comp., 1997].
We use our ideas to make progress on several related questions. We give satisfiability algorithms beating brute force search for depth- threshold circuits with a superlinear number of wires. These are the first such algorithms for depth greater than 2. We also show that Parity cannot be computed by polynomial-size AC^0 circuits with n^{o(1)} general threshold gates. Previously no lower bound for Parity in this setting could handle more than log(n) gates. This result also implies subexponential-time learning algorithms for AC^0 with n^{o(1)} threshold gates under the uniform distribution. In addition, we give almost optimal bounds for the number of gates in a depth-d threshold circuit computing Parity on average, and show average-case lower bounds for threshold formulas ofany depth.
Our techniques include adaptive random restrictions, anti-concentration and the structural theory of linear threshold functions, and bounded-read Chernoff bounds
Algorithms and lower bounds for de Morgan formulas of low-communication leaf gates
The class consists of Boolean functions
computable by size- de Morgan formulas whose leaves are any Boolean
functions from a class . We give lower bounds and (SAT, Learning,
and PRG) algorithms for , for classes
of functions with low communication complexity. Let
be the maximum -party NOF randomized communication
complexity of . We show:
(1) The Generalized Inner Product function cannot be computed in
on more than fraction of inputs
for As a corollary, we get an average-case lower bound for
against .
(2) There is a PRG of seed length that -fools . For
, we get the better seed length . This gives the first
non-trivial PRG (with seed length ) for intersections of half-spaces
in the regime where .
(3) There is a randomized -time SAT algorithm for , where In particular, this implies a nontrivial
#SAT algorithm for .
(4) The Minimum Circuit Size Problem is not in .
On the algorithmic side, we show that can be
PAC-learned in time
Quantified Derandomization of Linear Threshold Circuits
One of the prominent current challenges in complexity theory is the attempt
to prove lower bounds for , the class of constant-depth, polynomial-size
circuits with majority gates. Relying on the results of Williams (2013), an
appealing approach to prove such lower bounds is to construct a non-trivial
derandomization algorithm for . In this work we take a first step towards
the latter goal, by proving the first positive results regarding the
derandomization of circuits of depth .
Our first main result is a quantified derandomization algorithm for
circuits with a super-linear number of wires. Specifically, we construct an
algorithm that gets as input a circuit over input bits with
depth and wires, runs in almost-polynomial-time, and
distinguishes between the case that rejects at most inputs
and the case that accepts at most inputs. In fact, our
algorithm works even when the circuit is a linear threshold circuit, rather
than just a circuit (i.e., is a circuit with linear threshold gates,
which are stronger than majority gates).
Our second main result is that even a modest improvement of our quantified
derandomization algorithm would yield a non-trivial algorithm for standard
derandomization of all of , and would consequently imply that
. Specifically, if there exists a quantified
derandomization algorithm that gets as input a circuit with depth
and wires (rather than wires), runs in time at
most , and distinguishes between the case that rejects at
most inputs and the case that accepts at most
inputs, then there exists an algorithm with running time
for standard derandomization of .Comment: Changes in this revision: An additional result (a PRG for quantified
derandomization of depth-2 LTF circuits); rewrite of some of the exposition;
minor correction
Bounded Depth Circuits with Weighted Symmetric Gates: Satisfiability, Lower Bounds and Compression
A Boolean function f:{0,1}^n -> {0,1} is weighted symmetric if there exist a function g: Z -> {0,1} and integers w_0, w_1, ..., w_n such that f(x_1, ...,x_n) = g(w_0+sum_{i=1}^n w_i x_i) holds.
In this paper, we present algorithms for the circuit satisfiability problem of bounded depth circuits with AND, OR, NOT gates and a limited number of weighted symmetric gates. Our algorithms run in time super-polynomially faster than 2^n even when the number of gates is super-polynomial and the maximum weight of symmetric gates is nearly exponential. With an additional trick, we give an algorithm for the maximum satisfiability problem that runs in time poly(n^t)*2^{n-n^{1/O(t)}} for instances with n variables, O(n^t) clauses and arbitrary weights. To the best of our knowledge, this is the first moderately exponential time algorithm even for Max 2SAT instances with arbitrary weights.
Through the analysis of our algorithms, we obtain average-case lower bounds and compression algorithms for such circuits and worst-case lower bounds for majority votes of such circuits, where all the lower bounds are against the generalized Andreev function. Our average-case lower bounds might be of independent interest in the sense that previous ones for similar circuits with arbitrary symmetric gates rely on communication complexity lower bounds while ours are based on the restriction method
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