Dagstuhl Seminar Proceedings, Volume 7411
Dagstuhl Seminar Proceedings
DagSemProc
https://www.dagstuhl.de/dagpub/1862-4405
https://dblp.org/db/series/dagstuhl
1862-4405
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
7411
2008
https://drops.dagstuhl.de/entities/volume/DagSemProc-volume-7411
07411 Abstracts Collection – Algebraic Methods in Computational Complexity
From 07.10. to 12.10., the Dagstuhl Seminar 07411 ``Algebraic Methods in Computational Complexity'' was held in the International Conference and Research Center (IBFI),
Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and 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 paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.
Computational complexity
algebra
quantum computing
(de-) randomization
1-13
Regular Paper
Manindra
Agrawal
Manindra Agrawal
Harry
Buhrman
Harry Buhrman
Lance
Fortnow
Lance Fortnow
Thomas
Thierauf
Thomas Thierauf
10.4230/DagSemProc.07411.1
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07411 Executive Summary – Algebraic Methods in Computational Complexity
The seminar brought together almost 50 researchers covering a wide
spectrum of complexity theory. The focus on algebraic methods showed
once again the great importance of algebraic techniques for
theoretical computer science. We had almost 30 talks of length
between 15 and 45 minutes. This left enough room for discussions. We
had an open problem session that was very much appreciated.
Computational complexity
algebra
quantum computing
(de-) randomization
1-3
Regular Paper
Manindra
Agrawal
Manindra Agrawal
Harry
Buhrman
Harry Buhrman
Lance
Fortnow
Lance Fortnow
Thomas
Thierauf
Thomas Thierauf
10.4230/DagSemProc.07411.2
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Classical Simulation Complexity of Quantum Branching Programs
We present classical simulation techniques for measure once quantum
branching programs.
For bounded error syntactic quantum branching program of width $w$
that computes a function with error $delta$ we present a classical
deterministic branching program of the same length and width at most
$(1+2/(1-2delta))^{2w}$ that computes the same function.
Second technique is a classical stochastic simulation technique for
bounded error and unbounded error quantum branching programs. Our
result is that it is possible stochastically-classically simulate
quantum branching programs with the same length and almost the same
width, but we lost bounded error acceptance property.
Quantum algorithms
Branching Programs
Complexity
1-10
Regular Paper
Farid
Ablayev
Farid Ablayev
10.4230/DagSemProc.07411.3
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Diagonal Circuit Identity Testing and Lower Bounds
In this talk we give a deterministic polynomial time algorithm for testing whether a {em diagonal}
depth-$3$ circuit $C(arg{x}{n})$ (i.e. $C$ is a sum of powers of linear functions) is
zero.
Arithmetic circuit
identity testing
depth 3
depth 4
determinant
permanent
lower bound
1-1
Regular Paper
Nitin
Saxena
Nitin Saxena
10.4230/DagSemProc.07411.4
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High Entropy Random Selection Protocols
We study the two party problem of randomly selecting a string among
all the strings of length n. We want the protocol to have the
property that the output distribution has high entropy, even
when one of the two parties is dishonest and deviates from the
protocol. We develop protocols that achieve high, close to n,
entropy.
In the literature the randomness guarantee is usually expressed as
being close to the uniform distribution or in terms of resiliency.
The notion of entropy is not directly comparable to that of
resiliency, but we establish a connection between the two that
allows us to compare our protocols with the existing ones.
We construct an
explicit protocol that yields entropy n - O(1) and has 4log^* n
rounds, improving over the protocol of Goldwasser
et al. that also achieves this entropy but needs O(n)
rounds. Both these protocols need O(n^2) bits of communication.
Next we reduce the communication in our protocols. We show the existence,
non-explicitly, of a protocol that has 6-rounds, 2n + 8log n bits
of communication and yields entropy n- O(log n) and min-entropy
n/2 - O(log n). Our protocol achieves the same entropy bound as
the recent, also non-explicit, protocol of Gradwohl
et al., however achieves much higher min-entropy: n/2 -
O(log n) versus O(log n).
Finally we exhibit very simple explicit protocols. We connect the
security parameter of these geometric protocols with the well
studied Kakeya problem motivated by harmonic analysis and analytical
number theory. We are only able to prove that these protocols have
entropy 3n/4 but still n/2 - O(log n) min-entropy. Therefore
they do not perform as well with respect to the explicit
constructions of Gradwohl et al. entropy-wise, but still
have much better min-entropy. We conjecture that these simple
protocols achieve n -o(n) entropy. Our geometric
construction and its relation to the Kakeya problem follows a new and
different approach to the random selection problem than any of the
previously known protocols.
Shannon entropy
Random string ds
1-0
Regular Paper
Nikolai K.
Vereshchagin
Nikolai K. Vereshchagin
Harry
Buhrman
Harry Buhrman
Matthias
Cristandl
Matthias Cristandl
Michal
Koucky
Michal Koucky
Zvi
Lotker
Zvi Lotker
Boaz
Patt-Shamir
Boaz Patt-Shamir
10.4230/DagSemProc.07411.5
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The Unique Games Conjecture with Entangled Provers is False
We consider one-round games between a classical verifier and two provers who share entanglement. We show that
when the constraints enforced by the verifier are `unique' constraints (i.e., permutations), the value of the
game can be well approximated by a semidefinite program. Essentially the only algorithm known previously was
for the special case of binary answers, as follows from the work of Tsirelson in 1980. Among other things,
our result implies that the variant of the unique games conjecture where we allow the provers to share
entanglement is false. Our proof is based on a novel `quantum rounding technique', showing how to take a
solution to an SDP and transform it to a strategy for entangled provers.
Unique games
entanglement
1-17
Regular Paper
Julia
Kempe
Julia Kempe
Oded
Regev
Oded Regev
Ben
Toner
Ben Toner
10.4230/DagSemProc.07411.6
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Uniqueness of Optimal Mod 3 Circuits for Parity
We prove that the quadratic polynomials modulo $3$
with the largest correlation with parity are unique up to
permutation of variables and constant factors. As a consequence of
our result, we completely characterize the smallest
MAJ~$circ mbox{MOD}_3 circ {
m AND}_2$ circuits that compute parity, where a
MAJ~$circ mbox{MOD}_3 circ {
m AND}_2$ circuit is one that has a
majority gate as output, a middle layer of MOD$_3$ gates and a
bottom layer of AND gates of fan-in $2$. We
also prove that the sub-optimal circuits exhibit a stepped behavior:
any sub-optimal circuits of this class that compute parity
must have size at least a factor of $frac{2}{sqrt{3}}$ times the
optimal size. This verifies, for the special case of $m=3$,
two conjectures made
by Due~{n}ez, Miller, Roy and Straubing (Journal of Number Theory, 2006) for general MAJ~$circ mathrm{MOD}_m circ
{
m AND}_2$ circuits for any odd $m$. The correlation
and circuit bounds are obtained by studying the associated
exponential sums, based on some of the techniques developed
by Green (JCSS, 2004). We regard this as a step towards
obtaining tighter bounds both for the $m
ot = 3$ quadratic
case as well as for
higher degrees.
Circuit complexity
correlations
exponential sums
1-15
Regular Paper
Frederic
Green
Frederic Green
Amitabha
Roy
Amitabha Roy
10.4230/DagSemProc.07411.7
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