Theory of Computing
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Title : Special Issue: CCC 2018: Guest Editor's Foreword
Authors : Srikanth Srinivasan
Volume : 15
Number : 9
Pages : 1-3
URL : http://www.theoryofcomputing.org/articles/v015a009
Abstract
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Special Issue: CCC 2018
This collection comprises the expanded and fully refereed versions of
selected papers presented at the 33rd Computational Complexity
Conference (CCC 2018) held June 22-24, 2018 in San Diego, CA, USA.
These papers were selected by the Program Committee from among the
28 papers that appeared in the conference proceedings. Preliminary
versions of the papers were presented at the conference and the
extended abstracts appeared in the proceedings of the conference
published by Dagstuhl Publishing, LIPIcs. The CCC Program Committee
selected 28 out of 74 submissions for presentation at the conference;
of these, the four described below were invited to this Special Issue.
These four papers were refereed in accordance with the
rigorous standards of *Theory of Computing*.
* "Algebraic dependencies and PSPACE algorithms in approximative
complexity" by Zeyu Guo, Nitin Saxena, and Amit Sinhababu.
This paper makes contributions to two important problems in a
lgebraic complexity.
1. Algebraic independence: Given polynomials $f_1,...,f_m\in
\F[x_1,\ldots,x_n]$ as algebraic circuits, the question is to
check if they are algebraically independent. This notion is a
fundamental measure of the joint complexity of these polynomials.
In characteristic zero, this can be checked by a simple
polynomial-time (in terms of the sizes of the input circuits)
randomized algorithm via the well-known Jacobian criterion.
However, in positive characteristic, the best known result prior
to this result was $\NP^{\#P}$. In this paper, this problem is
shown to lie in the polynomial hierarchy (specifically, inside
$\AM\cap\coAM}$) for all characteristics.
2. Approximate polynomial satisfiability. This is a twist on the
Hilbert Nullstellensatz problem, which asks to check
if a given collection of multivariate polynomials $f_1,...,f_m\in
\F[x_1,\ldots,x_n]$ (given as algebraic circuits) has a common root
in the algebraic closure. Here, we are required to check if the
given collection of polynomials has an "approximate" root, or
equivalently, if the image of the polynomial map defined by
$(f_1,\ldots,f_m)$ contains $0$ in its Zariski closure. The
problem is motivated via connections to Geometric Complexity
Theory, where we try to understand the complexity of
"approximately" computing a given polynomial. In particular, the
authors show that a solution to Approximate Polynomial
Satisfiability implies the construction of explicit hitting sets
for $\overline{\VP},$ which is the class of polynomials
that can be approximated by polynomials in $\VP$.
The best known bound for this problem was previously
$\EXPSPACE$ via a Gröbner basis computation. This paper
places this problem in $\PSPACE$ over arbitrary fields. As
a corollary, this yields a $\PSPACE$ upper bound for
constructing explicit hitting sets for $\overline{\VP}$
over arbitrary fields, improving on previous results which
accomplished this in characteristic $0$.
* "Pseudorandom Generators from Polarizing Random Walks" by Eshan
Chattopadhyay, Pooya Hatami, Kaave Hosseini, and Shachar Lovett.
This paper introduces a new paradigm for constructing pseudorandom
generators (PRGs) for concrete computational models. This involves a
weaker variant of PRGs called fractional PRGs (fPRGs, first defined
in this paper) and it is shown that for Boolean function classes
closed under restrictions, explicit fPRGs can be converted to explicit
PRGs while only slightly weakening the parameters. In particular, this
yields a unified and simpler construction for PRGs for various
classes of functions that are either restricted in terms of their
Fourier spectra or simplify considerably upon applying random
restrictions. In some cases (e.g., Boolean functions with low
sensitivity), this yields a significant quantitative improvement.
In others, this idea significantly simplifies the proof
(e.g., $\AC^0$), since the construction works under quite
general assumptions.
* "The Cayley Semigroup Membership Problem" by Lukas Fleischer.
This paper concerns the semigroup membership problem:
the input is a semigroup $S$, given as a multiplication table,
a subset $X$ of $S$, and an element $t\in S$, and the question
is if $t$ is generated by $X$. The problem is known to be
$\NL$-complete in general and even $\P$-complete if the input
is relaxed to be a general groupoid. Here, the author makes progress
on the long-standing open problem on the complexity of the problem
when $S$ is a group. It is shown that in this case (and also the case
when $S$ is a commutative semigroup), the problem actually can be placed
in the class $\qAC^0$ (i.e., the problem has constant-depth
quasipolynomial-size circuits made up of OR, AND, and NOT gates),
and hence cannot be hard for any class of functions that contains
the PARITY function. Previously, it was only known that if $S$ is a
commutative group, then the problem has polynomial-size $\log \log
n$-depth circuits.
* "On The Hardness of Approximate and Exact (Bichromatic) Maximum Inner Product" by Lijie Chen: This paper deals with fine-grained complexity, where a central theme is to understand why we have not been able to improve long-standing algorithms for some problems in $\mathrm{P}$. The questions in this paper consider exact and approximation algorithms for the problem Max-IP, where the input is two lists $A$ and $B$ of $n$ vectors each and the quantity to be computed (or approximated) is the maximum inner product between a vector in $A$ and a vector in $B$. This problem generalizes the well-known Orthogonal Vectors problem, whose complexity is closely related to the Strong Exponential Time Hypothesis (SETH).
Under SETH (or the weaker Orthogonal Vectors Conjecture), the paper
proves a range of improved hardness results for many variants of
Max-IP, where the vectors are allowed to have entries that are Boolean
(i.e. $0,1$), $\{-1,1\}$-valued, or arbitrary integers. These hardness
results use many ideas including connections to MA-communication
complexity (specifically a recently improved MA-communication protocol
for Inner Product) and new connections to Quantum communication
complexity. In the Boolean case, matching upper bounds are also proved.
Finally, the paper also improves the parameters of the best known
MA-communication protocol for the Inner Product problem.
I would like to thank the authors for their contributions, the CCC
program committee for their initial reviews, Dieter van Melkebeek and
Venkatesan Guruswami for their advice on matters related to CCC,
Laszloo Babai for his advice on matters related to Theory of Computing,
and the anonymous referees for their hard work. It was a pleasure to
edit this Special Issue for Theory of Computing.
October 21, 2019
Srikanth Srinivasan
Guest Editor
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CCC 2018 Program Committee
Eric Allender (Rutgers University)
Paul Beame (University of Washington)
Eric Blais (University of Waterloo)
Mark Braverman (Princeton University)
Michael A. Forbes (University of Illinois at Urbana-Champaign)
Shafi Goldwasser (Massachusetts Institute of Technology and
Weizmann Institute of Science)
_Rocco Servedio_ (Columbia University) (Chair)
Srikanth Srinivasan (Indian Institute of Technology Bombay)
Thomas Thierauf (Aalen University)
Madhur Tulsiani (Toyota Technological Institute at Chicago)
Henry Yuen (University of California, Berkeley and University of Toronto)
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