eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2008-12-11
8381
1
18
10.4230/DagSemProc.08381.1
article
08381 Abstracts Collection – Computational Complexity of Discrete Problems
Miltersen, Peter Bro
Reischuk, Rüdiger
Schnitger, Georg
van Melkebeek, Dieter
From the 14th of September to the 19th of September, the Dagstuhl Seminar
08381 ``Computational Complexity of Discrete Problems'' was held in Schloss Dagstuhl - Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work as well as open problems were discussed.
Abstracts of the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this report. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol08381/DagSemProc.08381.1/DagSemProc.08381.1.pdf
Computational complexity
discrete problems
Turing machines
circuits
proof complexity
pseudorandomness
derandomization
cryptography
computational learning
communication complexity
query complexity
hardness of approximation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2008-12-11
8381
1
7
10.4230/DagSemProc.08381.2
article
08381 Executive Summary – Computational Complexity of Discrete Problems
Miltersen, Peter Bro
Reischuk, Rüdiger
Schnitger, Georg
van Melkebeek, Dieter
Estimating the computational complexity of discrete problems constitutes one of the central and classical topics in the theory of computation. Mathematicians and computer scientists have long tried to classify natural families of Boolean relations according to fundamental complexity measures like time and space, both in the uniform and in the nonuniform setting. Several models of computation have been developed in order to capture the various capabilities of digital computing devices, including parallelism, randomness, and quantum interference.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol08381/DagSemProc.08381.2/DagSemProc.08381.2.pdf
Computational complexity
discrete problems
Turing machines
circuits
proof complexity
pseudorandomness
derandomization
cryptography
computational learning
communication complexity
query complexity
hardness of approximation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2008-12-11
8381
1
9
10.4230/DagSemProc.08381.3
article
Approximation norms and duality for communication complexity lower bounds
Lee, Troy
Shraibman, Adi
Abstract: We will discuss a general norm based framework for showing
lower bounds on communication complexity. An advantage of this approach is that one can use duality theory to obtain a lower bound quantity phrased as a
maximization problem, which can be more convenient to work with in showing lower bounds.
We discuss two applications of this approach.
1. The approximation rank of a matrix A is the minimum rank of a
matrix close to A in ell_infty norm. The logarithm of approximation rank lower bounds quantum communication complexity and is one of the most powerful techniques available, albeit difficult to compute in practice. We
show that an approximation norm known as gamma_2 is polynomially
related to approximation rank.
This results in a polynomial time algorithm to approximate
approximation rank, and also shows that the logarithm of approximation rank lower bounds quantum communication complexity even with entanglement which was previously not known.
2. By means of an approximation norm which lower bounds multiparty
number-on-the-forehead complexity, we show non-trivial lower bounds on the complexity of the disjointness function for up to c log log n players, c <1.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol08381/DagSemProc.08381.3/DagSemProc.08381.3.pdf
Communication complexity
lower bounds
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2008-12-11
8381
1
11
10.4230/DagSemProc.08381.4
article
Depth Reduction for Circuits with a Single Layer of Modular Counting Gates
Hansen, Kristoffer Arnsfelt
We consider the class of constant depth AND/OR circuits augmented with
a layer of modular counting gates at the bottom layer, i.e ${AC}^0 circ {MOD}_m$ circuits. We show that the following
holds for several types of gates $G$: by adding a gate of type $G$ at
the output, it is possible to obtain an equivalent randomized depth 2
circuit of quasipolynomial size consisting of a gate of type $G$ at
the output and a layer of modular counting gates, i.e $G circ {MOD}_m$ circuits. The types of gates $G$ we consider are modular
counting gates and threshold-style gates. For all of these, strong
lower bounds are known for (deterministic) $G circ {MOD}_m$
circuits.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol08381/DagSemProc.08381.4/DagSemProc.08381.4.pdf
Boolean Circuits
Randomized Polynomials
Fourier sums
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2008-12-11
8381
1
33
10.4230/DagSemProc.08381.5
article
Fast polynomial factorization and modular composition
Kedlaya, Kiran
Umans, Christopher
We obtain randomized algorithms for factoring degree $n$
univariate polynomials over $F_q$ requiring $O(n^{1.5 +
o(1)} log^{1+o(1)} q+ n^{1 + o(1)}log^{2+o(1)} q)$ bit operations.
When $log q < n$, this is asymptotically faster than the best previous algorithms (von zur Gathen & Shoup (1992) and Kaltofen & Shoup (1998)); for
$log q ge n$, it matches the asymptotic running time of the best
known algorithms.
The improvements come from new algorithms for modular composition
of degree $n$ univariate polynomials, which is the asymptotic
bottleneck in fast algorithms for factoring polynomials over
finite fields. The best previous algorithms for modular
composition use $O(n^{(omega + 1)/2})$ field operations, where
$omega$ is the exponent of matrix multiplication (Brent & Kung
(1978)), with a slight improvement in the exponent achieved by
employing fast rectangular matrix multiplication (Huang & Pan
(1997)).
We show that modular composition and multipoint evaluation of
multivariate polynomials are essentially equivalent, in the sense
that an algorithm for one achieving exponent $alpha$ implies an
algorithm for the other with exponent $alpha + o(1)$, and vice
versa. We then give two new algorithms that solve the problem
optimally (up to lower order terms): an algebraic algorithm for
fields of characteristic at most $n^{o(1)}$, and a
nonalgebraic algorithm that works in arbitrary characteristic.
The latter algorithm works by lifting to characteristic 0,
applying a small number of rounds of {em multimodular reduction},
and finishing with a small number of multidimensional FFTs. The
final evaluations are reconstructed using the Chinese Remainder
Theorem. As a bonus, this algorithm produces a very efficient data
structure supporting polynomial evaluation queries, which is of
independent interest.
Our algorithms use techniques which are commonly employed in
practice, so they may be competitive for real problem sizes. This
contrasts with all previous subquadratic algorithsm for these
problems, which rely on fast matrix multiplication.
This is joint work with Kiran Kedlaya.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol08381/DagSemProc.08381.5/DagSemProc.08381.5.pdf
Modular composition; polynomial factorization; multipoint evaluation; Chinese Remaindering
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2008-12-11
8381
1
0
10.4230/DagSemProc.08381.6
article
Understanding space in resolution: optimal lower bounds and exponential trade-offs
Ben-Sasson, Eli
Nordström, Jakob
We continue the study of tradeoffs between space and length of
resolution proofs and focus on two new results:
begin{enumerate}
item
We show that length and space in resolution are uncorrelated. This
is proved by exhibiting families of CNF formulas of size $O(n)$ that
have proofs of length $O(n)$ but require space $Omega(n / log n)$. Our
separation is the strongest possible since any proof of length $O(n)$
can always be transformed into a proof in space $O(n / log n)$, and
improves previous work reported in [Nordstr"{o}m 2006, Nordstr"{o}m and
H{aa}stad 2008].
item We prove a number of trade-off results for space in the range
from constant to $O(n / log n)$, most of them superpolynomial or even
exponential. This is a dramatic improvement over previous results in
[Ben-Sasson 2002, Hertel and Pitassi 2007, Nordstr"{o}m 2007].
end{enumerate}
The key to our results is the following, somewhat surprising, theorem:
Any CNF formula $F$ can be transformed by simple substitution
transformation into a new formula $F'$ such that if $F$ has the right
properties, $F'$ can be proven in resolution in essentially the same
length as $F$ but the minimal space needed for $F'$ is lower-bounded
by the number of variables that have to be mentioned simultaneously in
any proof for $F$. Applying this theorem to so-called pebbling
formulas defined in terms of pebble games over directed acyclic graphs
and analyzing black-white pebbling on these graphs yields our results.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol08381/DagSemProc.08381.6/DagSemProc.08381.6.pdf
Proof complexity
Resolution
Pebbling.