306 research outputs found
Separating NOF communication complexity classes RP and NP
We provide a non-explicit separation of the number-on-forehead communication
complexity classes RP and NP when the number of players is up to \delta log(n)
for any \delta<1. Recent lower bounds on Set-Disjointness [LS08,CA08] provide
an explicit separation between these classes when the number of players is only
up to o(loglog(n))
A Nearly Optimal Lower Bound on the Approximate Degree of AC
The approximate degree of a Boolean function is the least degree of a real polynomial that
approximates pointwise to error at most . We introduce a generic
method for increasing the approximate degree of a given function, while
preserving its computability by constant-depth circuits.
Specifically, we show how to transform any Boolean function with
approximate degree into a function on variables with approximate degree at least . In particular, if , then
is polynomially larger than . Moreover, if is computed by a
polynomial-size Boolean circuit of constant depth, then so is .
By recursively applying our transformation, for any constant we
exhibit an AC function of approximate degree . This
improves over the best previous lower bound of due to
Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of
that holds for any function. Our lower bounds also apply to
(quasipolynomial-size) DNFs of polylogarithmic width.
We describe several applications of these results. We give:
* For any constant , an lower bound on the
quantum communication complexity of a function in AC.
* A Boolean function with approximate degree at least ,
where is the certificate complexity of . This separation is optimal
up to the term in the exponent.
* Improved secret sharing schemes with reconstruction procedures in AC.Comment: 40 pages, 1 figur
On the Complexity of Grid Coloring
This thesis studies problems at the intersection of Ramsey-theoretic mathematics, computational complexity, and communication complexity. The prototypical example of such a problem is Monochromatic-Rectangle-Free Grid Coloring. In an instance of Monochromatic-Rectangle-Free Grid Coloring, we are given a chessboard-like grid graph of dimensions n and m, where the vertices of the graph correspond to squares in the chessboard, and a number of allowed colors, c. The goal is to assign one of the allowed colors to each vertex of the grid graph so that no four vertices arranged in an axis-parallel rectangle are colored monochromatically. Our results include: 1. A conditional, graph-theoretic proof that deciding Monochromatic-Rectangle-Free Grid Coloring requires time superpolynomial in the input size. 2. A natural interpretation of Monochromatic-Rectangle-Free Grid Coloring as a lower bound on the communication complexity of a cluster of related predicates. 3. Original, best-yet, monochromatic-square-free grid colorings: a 2-coloring of the 13 x 13 grid, and a 3-coloring of the 39 x 39 grid. 4. An empirically-validated computational plan to decide a particular instance of Monochromatic-Rectangle-Free Grid Coloring that has been heavily studied by the broader theory community, but remains unsolved: whether the 17 x 17 grid can be 4-colored without monochromatic rectangles. Our plan is based in high-performance computing and is expected to take one year to complete
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
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
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