Dagstuhl Seminar Proceedings, Volume 9421
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
9421
2010
https://drops.dagstuhl.de/entities/volume/DagSemProc-volume-9421
09421 Abstracts Collection – Algebraic Methods in Computational Complexity
From 11.10. to 16.10.2009, the Dagstuhl Seminar 09421 ``Algebraic Methods in Computational Complexity '' was held in Schloss Dagstuhl~--~Leibniz Center for Informatics.
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
1-22
Regular Paper
Manindra
Agrawal
Manindra Agrawal
Lance
Fortnow
Lance Fortnow
Thomas
Thierauf
Thomas Thierauf
Christopher
Umans
Christopher Umans
10.4230/DagSemProc.09421.1
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09421 Executive Summary – Algebraic Methods in Computational Complexity
The seminar brought together more than 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, most of them about 40 minutes leaving ample room for
discussions. We also had a much appreciated open problem
session.
The talks ranged over a
broad assortment of subjects with the underlying theme of using
algebraic techniques. It was very fruitful and has hopefully
initiated new directions in research. Several participants
specifically mentioned that they appreciated the particular
focus on a common class of techniques (rather than end
results) as a unifying theme of the workshop. We look forward
to our next meeting!
Computational Complexity
Algebra
1-4
Regular Paper
Manindra
Agrawal
Manindra Agrawal
Lance
Fortnow
Lance Fortnow
Thomas
Thierauf
Thomas Thierauf
Christopher
Umans
Christopher Umans
10.4230/DagSemProc.09421.2
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An Axiomatic Approach to Algebrization
Non-relativization of complexity issues can be interpreted
as giving some evidence that these issues cannot be resolved
by "black-box" techniques. In the early 1990's, a sequence of
important non-relativizing results was proved, mainly using
algebraic techniques. Two approaches have been proposed
to understand the power and limitations of these algebraic
techniques: (1) Fortnow gives a construction of a class
of oracles which have a similar algebraic and logical structure,
although they are arbitrarily powerful. He shows that
many of the non-relativizing results proved using algebraic
techniques hold for all such oracles, but he does not show,
e.g., that the outcome of the "P vs. NP" question differs
between different oracles in that class. (2) Aaronson and
Wigderson give definitions of algebrizing separations and
collapses of complexity classes, by comparing classes relative
to one oracle to classes relative to an algebraic extension of
that oracle. Using these definitions, they show both that
the standard collapses and separations "algebrize" and that
many of the open questions in complexity fail to "algebrize",
suggesting that the arithmetization technique is close to its
limits. However, it is unclear how to formalize algebrization
of more complicated complexity statements than collapses
or separations, and whether the algebrizing statements are,
e.g., closed under modus ponens; so it is conceivable that
several algebrizing premises could imply (in a relativizing
way) a non-algebrizing conclusion.
Here, building on the work of Arora, Impagliazzo,
and Vazirani [4], we propose an axiomatic approach to "algebrization",
which complements and clarifies the approaches
of Fortnow and Aaronso&Wigderson. We present logical theories formalizing the notion of algebrizing techniques so that most algebrizing results
are provable within our theories and separations requiring
non-algebrizing techniques are independent of them.
Our theories extend the [AIV] theory formalizing relativization
by adding an Arithmetic Checkability axiom.
We show the following: (i) Arithmetic checkability holds
relative to arbitrarily powerful oracles (since Fortnow's algebraic oracles all satisfy Arithmetic Checkability
axiom); by contrast, Local Checkability of [AIV] restricts the
oracle power to NP cap co-NP. (ii) Most of the algebrizing
collapses and separations from [AW], such as IP = PSPACE,
NP subset ZKIP if one-way functions exist, MA-EXP not in P/poly,
etc., are provable from Arithmetic Checkability. (iii) Many
of the open complexity questions (shown to require nonalgebrizing
techniques in [AW]), such as "P vs. NP", "NP vs.
BPP", etc., cannot be proved from Arithmetic Checkability.
(iv) Arithmetic Checkability is also insufficient to prove one
known result, NEXP = MIP.
Oracles
arithmetization
algebrization
1-19
Regular Paper
Russell
Impagliazzo
Russell Impagliazzo
Valentine
Kabanets
Valentine Kabanets
Antonina
Kolokolova
Antonina Kolokolova
10.4230/DagSemProc.09421.3
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Deterministic approximation algorithms for the nearest codeword problem
The Nearest Codeword Problem (NCP) is a basic algorithmic question in the theory of error-correcting codes. Given a point v in F_2^n and a linear space L in F_2^n of dimension k NCP asks to find a point l in L that minimizes the (Hamming) distance from v.
It is well-known that the nearest codeword problem is NP-hard. Therefore approximation algorithms are of interest. The best effcient approximation algorithms for the NCP to date are due to Berman and Karpinski. They are a
deterministic algorithm that achieves an approximation ratio of O(k/c)
for an arbitrary constant c; and a randomized algorithm that achieves
an approximation ratio of O(k/ log n).
In this paper we present new deterministic algorithms for approximating
the NCP that improve substantially upon the earlier work, (almost) de-randomizing the randomized algorithm of Berman and Karpinski.
We also initiate a study of the following Remote Point Problem (RPP). Given a linear space L in F_2^n of dimension k RPP asks to find a point v in F_2^n that is far from L. We say that an algorithm achieves a remoteness of r for the RPP if it always outputs a point v that is at least r-far from L. In this paper we present a deterministic polynomial time algorithm that achieves a remoteness of Omega(n log k / k) for all k < n/2.
We motivate the remote point problem by relating it to both the nearest codeword problem and the matrix rigidity approach to circuit lower bounds in
computational complexity theory.
1-13
Regular Paper
Noga
Alon
Noga Alon
Rina
Panigrahy
Rina Panigrahy
Sergey
Yekhanin
Sergey Yekhanin
10.4230/DagSemProc.09421.4
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Learning Parities in the Mistake-Bound model
We study the problem of learning parity functions that depend on at most $k$ variables ($k$-parities) attribute-efficiently in the mistake-bound model.
We design a simple, deterministic, polynomial-time algorithm for learning $k$-parities with mistake bound $O(n^{1-frac{c}{k}})$, for any constant $c > 0$. This is the first polynomial-time algorithms that learns $omega(1)$-parities in the mistake-bound model with mistake bound $o(n)$.
Using the standard conversion techniques from the mistake-bound model to the PAC model, our algorithm can also be used for learning $k$-parities in the PAC model. In particular, this implies a slight improvement on the results of Klivans and Servedio
cite{rocco} for learning $k$-parities in the PAC model.
We also show that the $widetilde{O}(n^{k/2})$ time algorithm from cite{rocco} that PAC-learns $k$-parities with optimal sample complexity can be extended to the mistake-bound model.
Attribute-efficient learning
parities
mistake-bound
1-9
Regular Paper
Harry
Buhrman
Harry Buhrman
David
Garcia-Soriano
David Garcia-Soriano
Arie
Matsliah
Arie Matsliah
10.4230/DagSemProc.09421.5
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Planar Graph Isomorphism is in Log-Space
Graph Isomorphism is the prime example of a computational problem with a wide
difference between the best known lower and upper bounds on its complexity. There
is a significant gap between extant lower and upper bounds for planar graphs as well.
We bridge the gap for this natural and important special case by presenting an upper
bound that matches the known log-space hardness [JKMT03]. In fact, we show the
formally stronger result that planar graph canonization is in log-space. This improves the
previously known upper bound of AC1 [MR91].
Our algorithm first constructs the biconnected component tree of a connected planar
graph and then refines each biconnected component into a triconnected component
tree. The next step is to log-space reduce the biconnected planar graph isomorphism and
canonization problems to those for 3-connected planar graphs, which are known to be in
log-space by [DLN08]. This is achieved by using the above decomposition, and by making
significant modifications to Lindell’s algorithm for tree canonization, along with changes
in the space complexity analysis.
The reduction from the connected case to the biconnected case requires further new
ideas including a non-trivial case analysis and a group theoretic lemma to bound the
number of automorphisms of a colored 3-connected planar graph.
Planar Graphs
Graph Isomorphism
Logspace
1-32
Regular Paper
Samir
Datta
Samir Datta
Nutan
Limaye
Nutan Limaye
Prajakta
Nimbhorkar
Prajakta Nimbhorkar
Thomas
Thierauf
Thomas Thierauf
Fabian
Wagner
Fabian Wagner
10.4230/DagSemProc.09421.6
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Small space analogues of Valiant's classes and the limitations of skew formula
In the uniform circuit model of computation, the width of a boolean
circuit exactly characterises the ``space'' complexity of the
computed function. Looking for a similar relationship in Valiant's
algebraic model of computation, we propose width of an arithmetic
circuit as a possible measure of space. We introduce the class
VL as an algebraic variant of deterministic log-space L. In
the uniform setting, we show that our definition coincides with that
of VPSPACE at polynomial width.
Further, to define algebraic variants of non-deterministic
space-bounded classes, we introduce the notion of ``read-once''
certificates for arithmetic circuits. We show that polynomial-size
algebraic branching programs can be expressed as a read-once
exponential sum over polynomials in VL, ie
$mbox{VBP}inSigma^R cdotmbox{VL}$.
We also show that $Sigma^R cdot mbox{VBP} =mbox{VBP}$, ie
VBPs are stable under read-once exponential sums. Further, we
show that read-once exponential sums over a restricted class of
constant-width arithmetic circuits are within VQP, and this is the
largest known such subclass of poly-log-width circuits with this
property.
We also study the power of skew formulas and show that exponential
sums of a skew formula cannot represent the determinant polynomial.
Algebraic circuits
space bounds
circuit width
nondeterministic circuits
skew formulas
1-23
Regular Paper
Meena
Mahajan
Meena Mahajan
Raghavendra
Rao B. V.
Raghavendra Rao B. V.
10.4230/DagSemProc.09421.7
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Unconditional Lower Bounds against Advice
We show several unconditional lower bounds for exponential time classes against polynomial time classes with advice, including: (1) For any constant c, NEXP not in P^{NP[n^c]} (2) For any constant c, MAEXP not in MA/n^c (3) BPEXP not in BPP/n^{o(1)}.
It was previously unknown even whether NEXP in NP/n^{0.01}. For the probabilistic classes, no lower bounds for uniform exponential time against advice were known before. We also consider the question of whether these lower bounds can be made to work on almost all input lengths rather than on infinitely many. We give an oracle relative to which NEXP in i.o.NP, which provides evidence that this is not possible with current techniques.
Advice
derandomization
diagonalization
lower bounds
semantic classes
1-11
Regular Paper
Harry
Buhrman
Harry Buhrman
Lance
Fortnow
Lance Fortnow
Rahul
Santhanam
Rahul Santhanam
10.4230/DagSemProc.09421.8
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