Electronic Colloquium on Computational Complexity
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REPORTS > 2004:
All reports in year 2004:
TR04-001 | 11th December 2003
Lance Fortnow, Russell Impagliazzo, Chris Umans

On the complexity of succinct zero-sum games

We study the complexity of solving succinct zero-sum games,
i.e., the
games whose payoff matrix $M$ is given implicitly by a Boolean circuit
$C$ such that $M(i,j)=C(i,j)$. We complement the known $\EXP$-hardness
of computing the \emph{exact} value of a succinct zero-sum game by
several results on \emph{approximating} the value. (1) ... more >>>

TR04-002 | 8th January 2004
Troy Lee, Dieter van Melkebeek, Harry Buhrman

Language Compression and Pseudorandom Generators

The language compression problem asks for succinct descriptions of
the strings in a language A such that the strings can be efficiently
recovered from their description when given a membership oracle for
A. We study randomized and nondeterministic decompression schemes
and investigate how close we can get to the information ... more >>>

TR04-003 | 22nd December 2003
Pascal Koiran

Valiant's model and the cost of computing integers

Let $\tau(k)$ be the minimum number of arithmetic operations
required to build the integer $k \in \N$ from the constant 1.
A sequence $x_k$ is said to be ``easy to compute'' if
there exists a polynomial $p$ such that $\tau(x_k) \leq p(\log k)$
for all $k \geq ... more >>>

TR04-004 | 13th January 2004
Ramamohan Paturi, Pavel Pudlak

Circuit lower bounds and linear codes

In 1977 Valiant proposed a graph theoretical method for proving lower
bounds on algebraic circuits with gates computing linear functions.
He used this method to reduce the problem of proving
lower bounds on circuits with linear gates to to proving lower bounds
on the rigidity of a matrix, a ... more >>>

TR04-005 | 19th January 2004
Stasys Jukna

On Graph Complexity

Revisions: 1 , Comments: 1

A boolean circuit $f(x_1,\ldots,x_n)$ \emph{represents} a graph $G$
on $n$ vertices if for every input vector $a\in\{0,1\}^n$ with
precisely two $1$'s in, say, positions $i$ and $j$, $f(a)=1$
precisely when $i$ and $j$ are adjacent in $G$; on inputs with more
or less than two ... more >>>

TR04-006 | 6th January 2004
Günter Hotz

A remark on nondecidabilities of the initial value problem of ODEs

We prove that it is not decidable on R-machines if for a fixed finite intervall [a,b) the solution of the initial value problems of systems of ordinary differetial equations have solutions over this interval. This result holds independly from assumptions about differentiability of the right sides of the ODEs. Futhermore ... more >>>

TR04-007 | 13th January 2004
Alan L. Selman, Samik Sengupta

Polylogarithmic-round Interactive Proofs for coNP Collapses the Exponential Hierarchy

Revisions: 1 , Comments: 1

It is known \cite{BHZ87} that if every language in coNP has a
constant-round interactive proof system, then the polynomial hierarchy
collapses. On the other hand, Lund {\em et al}.\ \cite{LFKN92} have shown that
#SAT, the #P-complete function that outputs the number of satisfying
assignments of a Boolean ... more >>>

TR04-008 | 27th November 2003
Vikraman Arvind, Jacobo Toran

Solvable Group Isomorphism is (almost) in NP\cap coNP

The Group Isomorphism problem consists in deciding whether two input
groups $G_1$ and $G_2$ given by their multiplication tables are
isomorphic. We first give a 2-round Arthur-Merlin protocol for the
Group Non-Isomorphism problem such that on input groups $(G_1,G_2)$
of size $n$, Arthur uses ... more >>>

TR04-009 | 22nd January 2004
Martin Dyer, Alan Frieze, Thomas P. Hayes, Eric Vigoda

Randomly coloring constant degree graphs

We study a simple Markov chain, known as the Glauber dynamics, for generating a random <i>k</i>-coloring of a <i>n</i>-vertex graph with maximum degree &Delta;. We prove that the dynamics converges to a random coloring after <i>O</i>(<i>n</i> log <i>n</i>) steps assuming <i>k</i> &ge; <i>k</i><sub>0</sub> for some absolute constant <i>k</i><sub>0</sub>, and either: ... more >>>

TR04-010 | 26th January 2004
Michal Parnas, Dana Ron, Ronitt Rubinfeld

Tolerant Property Testing and Distance Approximation

A standard property testing algorithm is required to determine
with high probability whether a given object has property
P or whether it is \epsilon-far from having P, for any given
distance parameter \epsilon. An object is said to be \epsilon-far
from having ... more >>>

TR04-011 | 16th January 2004
Christian Glaßer

Counting with Counterfree Automata

We study the power of balanced regular leaf-languages.
First, we investigate (i) regular languages that are
polylog-time reducible to languages in dot-depth 1/2 and
(ii) regular languages that are polylog-time decidable.
For both classes we provide

- forbidden-pattern characterizations, and
- characterizations in terms of regular expressions.

Both ... more >>>

TR04-012 | 19th December 2003
Paul Beame, Joseph Culberson, David Mitchell, Cristopher Moore

The Resolution Complexity of Random Graph $k$-Colorability

We consider the resolution proof complexity of propositional formulas which encode random instances of graph $k$-colorability. We obtain a tradeoff between the graph density and the resolution proof complexity.
For random graphs with linearly many edges we obtain linear-exponential lower bounds on the length of resolution refutations. For any $\epsilon>0$, ... more >>>

TR04-013 | 10th February 2004
Oded Goldreich, Dana Ron

On Estimating the Average Degree of a Graph.

Following Feige, we consider the problem of
estimating the average degree of a graph.
Using ``neighbor queries'' as well as ``degree queries'',
we show that the average degree can be approximated
arbitrarily well in sublinear time, unless the graph is extremely sparse
(e.g., unless the graph has a sublinear ... more >>>

TR04-014 | 26th November 2003
Chris Pollett

Languages to diagonalize against advice classes

Variants of Kannan's Theorem are given where the circuits of
the original theorem are replaced by arbitrary recursively presentable
classes of languages that use advice strings and satisfy certain mild
conditions. These variants imply that $\DTIME(n^{k'})^{\NE}/n^k$
does not contain $\PTIME^{\NE}$, $\DTIME(2^{n^{k'}})/n^k$ does
not contain $\EXP$, $\SPACE(n^{k'})/n^k$ does not ... more >>>

TR04-015 | 24th February 2004
Richard Beigel, Harry Buhrman, Peter Fejer, Lance Fortnow, Piotr Grabowski, Luc Longpré, Andrej Muchnik, Frank Stephan, Leen Torenvliet

Enumerations of the Kolmogorov Function

A recursive enumerator for a function $h$ is an algorithm $f$ which
enumerates for an input $x$ finitely many elements including $h(x)$.
$f$ is an $k(n)$-enumerator if for every input $x$ of length $n$, $h(x)$
is among the first $k(n)$ elements enumerated by $f$.
If there is a $k(n)$-enumerator for ... more >>>

TR04-016 | 3rd March 2004
Michael Alekhnovich, Eli Ben-Sasson

Linear Upper Bounds for Random Walk on Small Density Random 3CNFs

We analyze the efficiency of the random walk algorithm on random 3CNF instances, and prove em linear upper bounds on the running time
of this algorithm for small clause density, less than 1.63. Our upper bound matches the observed running time to within a multiplicative factor. This is the ... more >>>

TR04-017 | 22nd February 2004
Evgeny Dantsin, Alexander Wolpert

Derandomization of Schuler's Algorithm for SAT

Recently Schuler \cite{Sch03} presented a randomized algorithm that
solves SAT in expected time at most $2^{n(1-1/\log_2(2m))}$ up to a
polynomial factor, where $n$ and $m$ are, respectively, the number of
variables and the number of clauses in the input formula. This bound
is the best known ... more >>>

TR04-018 | 24th January 2004
Jan Krajicek

Diagonalization in proof complexity

We study the diagonalization in the context of proof
complexity. We prove that at least one of the
following three conjectures is true:

1. There is a boolean function computable in E
that has circuit complexity $2^{\Omega(n)}$.

2. NP is not closed under the complement.

3. There is no ... more >>>

TR04-019 | 15th January 2004
Christian Glaßer, A. Pavan, Alan L. Selman, Samik Sengupta

Properties of NP-Complete Sets

We study several properties of sets that are complete for NP.
We prove that if $L$ is an NP-complete set and $S \not\supseteq L$ is a p-selective sparse set, then $L - S$ is many-one-hard for NP. We demonstrate existence of a sparse set $S \in \mathrm{DTIME}(2^{2^{n}})$
such ... more >>>

TR04-020 | 8th March 2004
Emanuele Viola

The Complexity of Constructing Pseudorandom Generators from Hard Functions

We study the complexity of building
pseudorandom generators (PRGs) from hard functions.

We show that, starting from a function f : {0,1}^l -> {0,1} that
is mildly hard on average, i.e. every circuit of size 2^Omega(l)
fails to compute f on at least a 1/poly(l)
fraction of inputs, we can ... more >>>

TR04-021 | 23rd March 2004
Eli Ben-Sasson, Oded Goldreich, Prahladh Harsha, Madhu Sudan, Salil Vadhan

Robust PCPs of Proximity, Shorter PCPs and Applications to Coding

We continue the study of the trade-off between the length of PCPs
and their query complexity, establishing the following main results
(which refer to proofs of satisfiability of circuits of size $n$):
We present PCPs of length $\exp(\tildeO(\log\log n)^2)\cdot n$
that can be verified by making $o(\log\log n)$ Boolean queries.
more >>>

TR04-022 | 31st March 2004
Nayantara Bhatnagar, Parikshit Gopalan

The Degree of Threshold Mod 6 and Diophantine Equations

We continue the study of the degree of polynomials representing threshold functions modulo 6 initiated by Barrington, Beigel and Rudich. We use the framework established by the authors relating representations by symmetric polynomials to simultaneous protocols. We show that proving bounds on the degree of Threshold functions is equivalent to ... more >>>

TR04-023 | 21st January 2004
Yaoyun Shi

Quantum and Classical Tradeoffs

We initiate the study of quantifying the quantumness of
a quantum circuit by the number of gates that do not preserve
the computational basis, as a means to understand the nature
of quantum algorithmic speedups.
Intuitively, a reduction in the quantumness requires
an increase in the amount of classical computation, ... more >>>

TR04-024 | 26th March 2004
Thomas Thierauf, Thanh Minh Hoang

On Closure Properties of GapL

Revisions: 1 , Comments: 1

We show necessary and sufficient conditions that
certain algebraic functions like the rank or the signature
of an integer matrix can be computed in GapL.

more >>>

TR04-025 | 24th January 2004
John Hitchcock, A. Pavan, Vinodchandran Variyam

Partial Bi-Immunity and NP-Completeness

The Turing and many-one completeness notions for $\NP$ have been
previously separated under {\em measure}, {\em genericity}, and {\em
bi-immunity} hypotheses on NP. The proofs of all these results rely
on the existence of a language in NP with almost everywhere hardness.

In this paper we separate the same NP-completeness ... more >>>

TR04-026 | 17th February 2004
Scott Aaronson

Limitations of Quantum Advice and One-Way Communication

Although a quantum state requires exponentially many classical bits to describe, the laws of quantum mechanics impose severe restrictions on how that state can be accessed. This paper shows in three settings that quantum messages have only limited advantages over classical ones.
First, we show that BQP/qpoly is contained in ... more >>>

TR04-027 | 20th February 2004
Henning Fernau

Parametric Duality: Kernel Sizes and Algorithmics

We derive the first lower bound results on kernel sizes of parameterized problems. The same idea also allows us to sometimes "de-parameterize"
parameterized algorithms.

more >>>

TR04-028 | 19th March 2004
Arfst Nickelsen, Till Tantau, Lorenz Weizsäcker

Aggregates with Component Size One Characterize Polynomial Space

Aggregates are a computational model similar to circuits, but the
underlying graph is not necessarily acyclic. Logspace-uniform
polynomial-size aggregates decide exactly the languages in PSPACE;
without uniformity condition they decide the languages in
PSPACE/poly. As a measure of similarity to boolean circuits we
introduce the parameter component size. We ... more >>>

TR04-029 | 7th April 2004
John Hitchcock, Maria Lopez-Valdes, Elvira Mayordomo

Scaled dimension and the Kolmogorov complexity of Turing hard sets

Scaled dimension has been introduced by Hitchcock et al (2003) in order to quantitatively distinguish among classes such as SIZE(2^{a n}) and SIZE(2^{n^{a}}) that have trivial dimension and measure in ESPACE.

more >>>

TR04-030 | 9th March 2004
Nikolay Vereshchagin

Kolmogorov complexity of enumerating finite sets

Solovay has proven that
the minimal length of a program enumerating a set A
is upper bounded by 3 times the absolute value of the
logarithm of the
probability that a random program will enumerate A.
It is unknown whether one can replace the constant
3 by a smaller constant.
more >>>

TR04-031 | 22nd March 2004
Troy Lee, Andrei Romashchenko

On Polynomially Time Bounded Symmetry of Information

The information contained in a string $x$ about a string $y$
is defined as the difference between the Kolmogorov complexity
of $y$ and the conditional Kolmogorov complexity of $y$ given $x$,
i.e., $I(x:y)=\C(y)-\C(y|x)$. From the well-known Kolmogorov--Levin Theorem it follows that $I(x:y)$ is symmetric up to a small ... more >>>

TR04-032 | 5th February 2004
Ryan Williams

A new algorithm for optimal constraint satisfaction and its implications

We present a novel method for exactly solving (in fact, counting solutions to) general constraint satisfaction optimization with at most two variables per constraint (e.g. MAX-2-CSP and MIN-2-CSP), which gives the first exponential improvement over the trivial algorithm; more precisely, it is a constant factor improvement in the base of ... more >>>

TR04-033 | 23rd January 2004
Michael Schmitt

On the sample complexity of learning for networks of spiking neurons with nonlinear synaptic interactions

We study networks of spiking neurons that use the timing of pulses
to encode information. Nonlinear interactions model the spatial
groupings of synapses on the dendrites and describe the computations
performed at local branches. We analyze the question of how many
examples these networks must ... more >>>

TR04-034 | 12th April 2004
April Rasala Lehman, Eric Lehman

Network Coding: Does the Model Need Tuning?

We consider the general network information flow problem, which was
introduced by Ahlswede et. al. We show a periodicity effect: for
every integer m greater than 1, there exists an instance of the
network information flow problem that admits a solution if and only if
the alphabet size is a ... more >>>

TR04-035 | 10th April 2004
Scott Contini, Ernie Croot, Igor E. Shparlinski

Complexity of Inverting the Euler Function

We present an algorithm to invert the Euler function $\varphi(m)$. The algorithm, for a given integer $n\ge 1$, in polynomial time ``on average'', finds theset $\Psi(n)$ of all solutions $m$ to the equation $\varphi(m) =n$. In fact, in the worst case the set $\Psi(n)$ is exponentially large and cannot be ... more >>>

TR04-036 | 27th March 2004
Ziv Bar-Yossef, T.S. Jayram, Iordanis Kerenidis

Exponential Separation of Quantum and Classical One-Way Communication Complexity

We give the first exponential separation between quantum and bounded-error randomized one-way communication complexity. Specifically, we define the Hidden Matching Problem HM_n: Alice gets as input a string x in {0,1}^n and Bob gets a perfect matching M on the n coordinates. Bob's goal is to output a tuple (i,j,b) ... more >>>

TR04-037 | 14th April 2004
Elmar Böhler, Christian Glaßer, Bernhard Schwarz, Klaus W. Wagner

Generation Problems

Given a fixed computable binary operation f, we study the complexity of the following generation problem: The input consists of strings a1,...,an,b. The question is whether b is in the closure of {a1,...,an} under operation f.

For several subclasses of operations we prove tight upper and lower bounds for the ... more >>>

TR04-038 | 27th April 2004
John Case, Sanjay Jain, Rüdiger Reischuk, Frank Stephan, Thomas Zeugmann

A Polynomial Time Learner for a Subclass of Regular Patterns

Presented is an algorithm (for learning a subclass of erasing regular
pattern languages) which
can be made to run with arbitrarily high probability of
success on extended regular languages generated by patterns
$\pi$ of the form $x_0 \alpha_1 x_1 ... \alpha_m x_m$
for unknown $m$ but known $c$,
more >>>

TR04-039 | 21st April 2004
Andrzej Lingas, Martin Wahlén

On approximation of the maximum clique minor containment problem and some subgraph homeomorphism problems

We consider the ``minor'' and ``homeomorphic'' analogues of the maximum clique problem, i.e., the problems of determining the largest $h$ such that the input graph has a minor isomorphic to $K_h$ or a subgraph homeomorphic to $K_h,$ respectively.We show the former to be approximable within $O(\sqrt {n} \log^{1.5} n)$ by ... more >>>

TR04-040 | 4th May 2004
Venkatesan Guruswami, Alexander Vardy

Maximum-likelihood decoding of Reed-Solomon codes is NP-hard

Maximum-likelihood decoding is one of the central algorithmic
problems in coding theory. It has been known for over 25 years
that maximum-likelihood decoding of general linear codes is
NP-hard. Nevertheless, it was so far unknown whether maximum-
likelihood decoding remains hard for any specific family of
more >>>

TR04-041 | 18th May 2004
Michael Alekhnovich, Edward Hirsch, Dmitry Itsykson

Exponential lower bounds for the running time of DPLL algorithms on satisfiable formulas

DPLL (for Davis, Putnam, Logemann, and Loveland) algorithms form the largest family of contemporary algorithms for SAT (the propositional satisfiability problem) and are widely used in applications. The recursion trees of DPLL algorithm executions on unsatisfiable formulas are equivalent to tree-like resolution proofs. Therefore, lower bounds for tree-like resolution (which ... more >>>

TR04-042 | 21st May 2004
Ran Raz

Multilinear-$NC_1$ $\ne$ Multilinear-$NC_2$

An arithmetic circuit or formula is multilinear if the polynomial
computed at each of its wires is multilinear.
We give an explicit example for a polynomial $f(x_1,...,x_n)$,
with coefficients in $\{0,1\}$, such that over any field:
1) $f$ can be computed by a polynomial-size multilinear circuit
of depth $O(\log^2 ... more >>>

TR04-043 | 20th May 2004
Luca Trevisan

Some Applications of Coding Theory in Computational Complexity

Error-correcting codes and related combinatorial constructs
play an important role in several recent (and old) results
in computational complexity theory. In this paper we survey
results on locally-testable and locally-decodable error-correcting
codes, and their applications to complexity theory and to

Locally decodable codes are error-correcting codes ... more >>>

TR04-044 | 1st June 2004
Eric Allender, Harry Buhrman, Michal Koucky

What Can be Efficiently Reduced to the Kolmogorov-Random Strings?

We investigate the question of whether one can characterize complexity
classes (such as PSPACE or NEXP) in terms of efficient
reducibility to the set of Kolmogorov-random strings R_C.
We show that this question cannot be posed without explicitly dealing
with issues raised by the choice of universal
machine in the ... more >>>

TR04-045 | 15th April 2004
Hartmut Klauck, Robert Spalek, Ronald de Wolf

Quantum and Classical Strong Direct Product Theorems and Optimal Time-Space Tradeoffs

A strong direct product theorem says that if we want to compute
k independent instances of a function, using less than k times
the resources needed for one instance, then our overall success
probability will be exponentially small in k.
We establish such theorems for the classical as well as ... more >>>

TR04-046 | 4th June 2004
Eli Ben-Sasson, Madhu Sudan

Robust Locally Testable Codes and Products of Codes

We continue the investigation of locally testable codes, i.e.,
error-correcting codes for whom membership of a given word in the
code can be tested probabilistically by examining it in very few
locations. We give two general results on local testability:
First, motivated by the recently proposed notion of robust
probabilistically ... more >>>

TR04-047 | 22nd April 2004
Xiaoyang Gu

A note on dimensions of polynomial size circuits

In this paper, we use resource-bounded dimension theory to investigate polynomial size circuits. We show that for every $i\geq 0$, $\Ppoly$ has $i$th order scaled $\pthree$-strong dimension $0$. We also show that $\Ppoly^\io$ has $\pthree$-dimension $1/2$, $\pthree$-strong dimension $1$. Our results improve previous measure results of Lutz (1992) and dimension ... more >>>

TR04-048 | 21st April 2004
André Lanka, Andreas Goerdt

An approximation hardness result for bipartite Clique

Assuming 3-SAT formulas are hard to refute
on average, Feige showed some approximation hardness
results for several problems like min bisection, dense
$k$-subgraph, max bipartite clique and the 2-catalog segmentation
problem. We show a similar result for
max bipartite clique, but under the assumption, 4-SAT formulas
are hard to refute ... more >>>

TR04-049 | 15th June 2004
Piotr Berman, Marek Karpinski, Yakov Nekrich

Optimal Trade-Off for Merkle Tree Traversal

We prove upper and lower bounds for computing Merkle tree
traversals, and display optimal trade-offs between time
and space complexity of that problem.

more >>>

TR04-050 | 13th June 2004
Michelle Effros, Leonard J. Schulman

Deterministic clustering with data nets

We consider the $K$-clustering problem with the $\ell_2^2$
distortion measure, also known as the problem of optimal
fixed-rate vector quantizer design. We provide a deterministic
approximation algorithm which works for all dimensions $d$ and
which, given a data set of size $n$, computes in time
more >>>

TR04-051 | 10th June 2004
Zdenek Dvorák, Daniel Král, Ondrej Pangrác

Locally consistent constraint satisfaction problems

An instance of a constraint satisfaction problem is $l$-consistent
if any $l$ constraints of it can be simultaneously satisfied.
For a set $\Pi$ of constraint types, $\rho_l(\Pi)$ denotes the largest ratio of constraints which can be satisfied in any $l$-consistent instance composed by constraints from the set $\Pi$. In the ... more >>>

TR04-052 | 14th June 2004
Michael Ben Or, Don Coppersmith, Michael Luby, Ronitt Rubinfeld

Non-Abelian Homomorphism Testing, and Distributions Close to their Self-Convolutions

In this paper, we study two questions related to
the problem of testing whether a function is close to a homomorphism.
For two finite groups $G,H$ (not necessarily Abelian),
an arbitrary map $f:G \rightarrow H$, and a parameter $0 < \epsilon <1$,
say that $f$ is $\epsilon$-close to a homomorphism ... more >>>

TR04-053 | 17th June 2004
A. Pavan, Vinodchandran Variyam

Polylogarithmic Round Arthur-Merlin Games and Random-Self-Reducibility

We consider Arthur-Merlin proof systems where (a) Arthur is a probabilistic quasi-polynomial time Turing machine
and (AMQ)(b) Arthur is a probabilistic exponential time Turing machine (AME). We prove two new results related to these classes.

more >>>

TR04-054 | 5th June 2004
Andrej Muchnik, Alexander Shen, Nikolay Vereshchagin, Mikhail V. Vyugin

Non-reducible descriptions for conditional Kolmogorov complexity

Let a program p on input a output b. We are looking for a
shorter program p' having the same property (p'(a)=b). In
addition, we want p' to be simple conditional to p (this
means that the conditional Kolmogorov complexity K(p'|p) is
negligible). In the present paper, we prove that ... more >>>

TR04-055 | 27th May 2004
Kousha Etessami, Andreas Lochbihler

The computational complexity of Evolutionarily Stable Strategies

Game theory has been used for a long time to study phenomena in evolutionary biology, beginning systematically with the seminal work of John Maynard Smith. A central concept in this connection has been the notion of an evolutionarily stable strategy (ESS) in a symmetric two-player strategic form game. A regular ... more >>>

TR04-056 | 1st July 2004
Vinodchandran Variyam

A note on the circuit complexity of PP

In this short note we show that for any integer k, there are
languages in the complexity class PP that do not have Boolean
circuits of size $n^k$.

more >>>

TR04-057 | 16th May 2004
Monica del Pilar Canales Chacon, Michael Johannes Vielhaber

Structural and Computational Complexity of Isometries and their Shift Commutators

{\bf Abstract}

Isometries on formal power series over the finite field $\ff_2$
or on $2$--adic integers can be
computed by invertible transducers on inputs from $\{0,1\}^\infty$.
We consider the structural complexity of an isometry $f$,
measured as {\it tree complexity} $T(f,h)$, $h$ the tree height
[H.~Niederreiter, M.~Vielhaber, {\it J.~Cpx.}, ... more >>>

TR04-058 | 28th May 2004
John Case, Sanjay Jain, Eric Martin, Arun Sharma, Frank Stephan

Identifying Clusters from Positive Data

The present work studies clustering from an abstract point of view
and investigates its properties in the framework of inductive inference.
Any class $S$ considered is given by a numbering
$A_0,A_1,...$ of nonempty subsets of the natural numbers
or the rational k-dimensional vector space as a hypothesis space.
A clustering ... more >>>

TR04-059 | 21st June 2004
Beatrice List, Markus Maucher, Uwe Schöning, Rainer Schuler

Randomized Quicksort and the Entropy of the Random Number Generator

The worst-case complexity of an implementation of Quicksort depends
on the random number generator that is used to select the pivot
elements. In this paper we estimate the expected number of
comparisons of Quicksort as a function in the entropy of the random
source. We give upper and lower bounds ... more >>>

TR04-060 | 22nd July 2004
Eli Ben-Sasson, Madhu Sudan

Simple PCPs with Poly-log Rate and Query Complexity

We give constructions of PCPs of length n*polylog(n) (with respect
to circuits of size n) that can be verified by making polylog(n)
queries to bits of the proof. These PCPs are not only shorter than
previous ones, but also simpler. Our (only) building blocks are
Reed-Solomon codes and the bivariate ... more >>>

TR04-061 | 30th June 2004
Scott Aaronson

The Complexity of Agreement

A celebrated 1976 theorem of Aumann asserts that honest, rational
Bayesian agents with common priors will never "agree to disagree": if
their opinions about any topic are common knowledge, then those
opinions must be equal. Economists have written numerous papers
examining the assumptions behind this theorem. But two key questions
more >>>

TR04-062 | 28th July 2004
Stasys Jukna

A note on the P versus NP intersected with co-NP question in communication complexity

Revisions: 1 , Comments: 1

We consider the P versus NP\cap coNP question for the classical two-party communication protocols: if both a boolean function and its negation have small nondeterministic communication complexity, what is then its deterministic and/or probabilistic communication complexity? In the fixed (worst) partition case this question was answered by Aho, Ullman and ... more >>>

TR04-063 | 23rd July 2004
Yehuda Lindell, Benny Pinkas

A Proof of Yao's Protocol for Secure Two-Party Computation

Revisions: 1

In the mid 1980's, Yao presented a constant-round protocol for
securely computing any two-party functionality in the presence of
semi-honest adversaries (FOCS 1986). In this paper, we provide a
complete description of Yao's protocol, along with a rigorous
proof of security. Despite the importance of Yao's protocol to the
field ... more >>>

TR04-064 | 25th June 2004
Piotr Faliszewski

Exponential time reductions and sparse languages in NEXP

In this paper we define a many-one reduction which is allowed to work in exponential time but may only output polynomially many symbols. We show that there are no NEXP-hard sparse languages under our reduction unless EXP=UEXP.

more >>>

TR04-065 | 28th July 2004
Luca Trevisan

Inapproximability of Combinatorial Optimization Problems

Revisions: 1

We survey results on the hardness of approximating combinatorial
optimization problems.

more >>>

TR04-066 | 6th July 2004
Tomoyuki Yamakami, Toshio Suzuki

Resource Bounded Immunity and Simplicity

Revisiting the thirty years-old notions of resource-bounded immunity and simplicity, we investigate the structural characteristics of various immunity notions: strong immunity, almost immunity, and hyperimmunity as well as their corresponding simplicity notions. We also study limited immunity and simplicity, called k-immunity and feasible k-immunity, and their simplicity notions. Finally, we ... more >>>

TR04-067 | 20th July 2004
hadi salmasian, ravindran kannan, Santosh Vempala

The Spectral Method for Mixture Models

We present an algorithm for learning a mixture of distributions.
The algorithm is based on spectral projection and
is efficient when the components of the mixture are logconcave

more >>>

TR04-068 | 13th August 2004
Nir Ailon, Bernard Chazelle

Information Theory in Property Testing and Monotonicity Testing in Higher Dimension

In general property testing, we are given oracle access to a function $f$, and we wish to randomly test if the function satisfies a given property $P$, or it is $\epsilon$-far from having that property. In a more general setting, the domain on which the function is defined is equipped ... more >>>

TR04-069 | 13th August 2004
Eran Rom, Amnon Ta-Shma

Improving the alphabet size in high noise, almost optimal rate list decodable codes

Revisions: 2

In this note we revisit the construction of high noise, almost
optimal rate list decodable code of Guruswami ("Better extractors for better codes?")
Guruswami showed that based on optimal extractors one can build a
$(1-\epsilon,O({1 \over \epsilon}))$ list decodable codes of rate
$\Omega({\epsilon \over {log{1 \over \epsilon}}})$ and alphabet
size ... more >>>

TR04-070 | 22nd June 2004
Leonid Gurvits

Combinatorial and algorithmic aspects of hyperbolic polynomials

Let $p(x_1,...,x_n) =\sum_{ (r_1,...,r_n) \in I_{n,n} } a_{(r_1,...,r_n) } \prod_{1 \leq i \leq n} x_{i}^{r_{i}}$
be homogeneous polynomial of degree $n$ in $n$ real variables with integer nonnegative coefficients.
The support of such polynomial $p(x_1,...,x_n)$
is defined as $supp(p) = \{(r_1,...,r_n) \in I_{n,n} : a_{(r_1,...,r_n)} \neq 0 ... more >>>

TR04-071 | 11th August 2004
Marcus Schaefer, Stephen A. Fenner

Simplicity and Strong Reductions

A set is called NP-simple if it lies in NP, and its complement is infinite, and does not contain any infinite subsets in NP. Hartmanis, Li and Yesha proved that no set which is hard for NP under many-one (Karp) reductions is NP-simple unless the intersection of NP and coNP ... more >>>

TR04-072 | 19th August 2004
John Hitchcock

Hausdorff Dimension and Oracle Constructions

Bennett and Gill (1981) proved that P^A != NP^A relative to a
random oracle A, or in other words, that the set
O_[P=NP] = { A | P^A = NP^A }
has Lebesgue measure 0. In contrast, we show that O_[P=NP] has
Hausdorff dimension 1.

... more >>>

TR04-073 | 9th July 2004
Henning Fernau

A Top-Down Approach to Search-Trees: Improved Algorithmics for 3-Hitting Set

In this paper, we show how to systematically
improve on parameterized algorithms and their
analysis, focusing on search-tree based algorithms
for d-Hitting Set, especially for d=3.
We concentrate on algorithms which are easy to implement,
in contrast with the highly sophisticated algorithms
which have been elsewhere designed to ... more >>>

TR04-074 | 26th August 2004
Emanuele Viola

On Parallel Pseudorandom Generators

Revisions: 1

We study pseudorandom generator (PRG) constructions $G^f : {0,1}^l \to {0,1}^{l+s}$ from one-way functions $f : {0,1}^n \to {0,1}^m$. We consider PRG constructions of the form $G^f(x) = C(f(q_{1}) \ldots f(q_{poly(n)}))$
where $C$ is a polynomial-size constant depth circuit
and $C$ and the $q$'s are generated from $x$ arbitrarily.
more >>>

TR04-075 | 21st July 2004
Michael Schmitt

Some dichotomy theorems for neural learning problems

The computational complexity of learning from binary examples is
investigated for linear threshold neurons. We introduce
combinatorial measures that create classes of infinitely many
learning problems with sample restrictions. We analyze how the
complexity of these problems depends on the values for the measures.
... more >>>

TR04-076 | 17th September 2004
Oliver Giel, Ingo Wegener

Searching Randomly for Maximum Matchings

Many real-world optimization problems in, e.g., engineering
or biology have the property that not much is known about
the function to be optimized. This excludes the application
of problem-specific algorithms. Simple randomized search
heuristics are then used with surprisingly good results. In
order to understand the working principles behind such
more >>>

TR04-077 | 17th July 2004
Alina Beygelzimer, Varsha Dani, Tom Hayes, John Langford

Reductions Between Classification Tasks

There are two approaches to solving a new supervised learning task: either
analyze the task independently or reduce it to a task that has already
been thoroughly analyzed. This paper investigates the latter approach for
classification problems. In addition to obvious theoretical motivations,
there is fairly strong empirical evidence that ... more >>>

TR04-078 | 3rd August 2004
Henning Fernau

Two-Layer Planarization: Improving on Parameterized Algorithmics

A bipartite graph is biplanar if the vertices can be
placed on two parallel lines in the plane such that there are
no edge crossings when edges are drawn as straight-line segments.
We study two variants of biplanarization problems:
- Two-Layer Planarization TLP: can $k$ edges be deleted from
a ... more >>>

TR04-079 | 30th August 2004
John Hitchcock, Jack H. Lutz, Sebastiaan Terwijn

The Arithmetical Complexity of Dimension and Randomness

Constructive dimension and constructive strong dimension are effectivizations of the Hausdorff and packing dimensions, respectively. Each infinite binary sequence A is assigned a dimension dim(A) in [0,1] and a strong dimension Dim(A) in [0,1].

Let DIM^alpha and DIMstr^alpha be the classes of all sequences of dimension alpha and of strong ... more >>>

TR04-080 | 7th September 2004
Lance Fortnow, Troy Lee, Nikolay Vereshchagin

Kolmogorov Complexity with Error

We introduce the study of Kolmogorov complexity with error. For a
metric d, we define C_a(x) to be the length of a shortest
program p which prints a string y such that d(x,y) \le a. We
also study a conditional version of this measure C_{a,b}(x|y)
where the task is, given ... more >>>

TR04-081 | 9th September 2004
Harry Buhrman, Lance Fortnow, Ilan Newman, Nikolay Vereshchagin

Increasing Kolmogorov Complexity

How much do we have to change a string to increase its Kolmogorov complexity. We show that we can
increase the complexity of any non-random string of length n by flipping O(sqrt(n)) bits and some strings
Omega(sqrt(n)) bit flips. For a given m, we also give bounds for ... more >>>

TR04-082 | 9th September 2004
Olaf Beyersdorff

Representable Disjoint NP-Pairs

Revisions: 1

We investigate the class of disjoint NP-pairs under different reductions.
The structure of this class is intimately linked to the simulation order
of propositional proof systems, and we make use of the relationship between
propositional proof systems and theories of bounded arithmetic as the main
tool of our analysis.
more >>>

TR04-083 | 8th September 2004
Boaz Barak, Yehuda Lindell, Salil Vadhan

Lower Bounds for Non-Black-Box Zero Knowledge

We show new lower bounds and impossibility results for general (possibly <i>non-black-box</i>) zero-knowledge proofs and arguments. Our main results are that, under reasonable complexity assumptions:
<li> There does not exist a two-round zero-knowledge <i>proof</i> system with perfect completeness for an NP-complete language. The previous impossibility result for two-round zero ... more >>>

TR04-084 | 28th September 2004
George Karakostas

A better approximation ratio for the Vertex Cover problem

We reduce the approximation factor for Vertex Cover to $2-\Theta(1/\sqrt{logn})$
(instead of the previous $2-\Theta(loglogn/logn})$, obtained by Bar-Yehuda and Even,
and by Monien and Speckenmeyer in 1985. The improvement of the vanishing
factor comes as an application of the recent results of Arora, Rao, and Vazirani
that improved ... more >>>

TR04-085 | 3rd October 2004
Erez Petrank, Guy Rothblum

Selection from Structured Data Sets

A large body of work studies the complexity of selecting the
$j$-th largest element in an arbitrary set of $n$ elements (a.k.a.
the select$(j)$ operation). In this work, we study the
complexity of select in data that is partially structured by
an initial preprocessing stage and in a data structure ... more >>>

TR04-086 | 12th October 2004
Ronen Shaltiel, Chris Umans

Pseudorandomness for Approximate Counting and Sampling

We study computational procedures that use both randomness and nondeterminism. Examples are Arthur-Merlin games and approximate counting and sampling of NP-witnesses. The goal of this paper is to derandomize such procedures under the weakest possible assumptions.

Our main technical contribution allows one to ``boost'' a given hardness assumption. One special ... more >>>

TR04-087 | 13th October 2004
Alexander Healy, Salil Vadhan, Emanuele Viola

Using Nondeterminism to Amplify Hardness

We revisit the problem of hardness amplification in $\NP$, as
recently studied by O'Donnell (STOC `02). We prove that if $\NP$
has a balanced function $f$ such that any circuit of size $s(n)$
fails to compute $f$ on a $1/\poly(n)$ fraction of inputs, then
$\NP$ has a function $f'$ such ... more >>>

TR04-088 | 12th October 2004
Emanuele Viola, Dan Gutfreund

Fooling Parity Tests with Parity Gates

We study the complexity of computing $k$-wise independent and
$\epsilon$-biased generators $G : \{0,1\}^n \to \{0,1\}^m$.
Specifically, we refer to the complexity of computing $G$ \emph{explicitly}, i.e.
given $x \in \{0,1\}^n$ and $i \in \{0,1\}^{\log m}$, computing the $i$-th output bit of $G(x)$.
Mansour, Nisan and Tiwari (1990) show that ... more >>>

TR04-089 | 26th October 2004
Ingo Wegener

Simulated Annealing Beats Metropolis in Combinatorial Optimization

The Metropolis algorithm is simulated annealing with a fixed temperature.Surprisingly enough, many problems cannot be solved more efficiently by simulated annealing than by the Metropolis algorithm with the best temperature. The problem of finding a natural example (artificial examples are known) where simulated annealing outperforms the Metropolis algorithm for all ... more >>>

TR04-090 | 3rd November 2004
Kazuyuki Amano, Akira Maruoka

Better Simulation of Exponential Threshold Weights by Polynomial Weights

We give an explicit construction of depth two threshold circuit with polynomial weights and $\tilde{O}(n^5)$ gates that computes an arbitrary threshold function. We also give the construction of such circuits with $O(n^3/\log n)$ gates computing the COMPARISON and CARRY functions, and that with $O(n^4/\log n)$ gates computing the ADDITION function. ... more >>>

TR04-091 | 29th September 2004
Ondrej Klíma, Pascal Tesson, Denis Thérien

Dichotomies in the Complexity of Solving Systems of Equations over Finite Semigroups

We consider the problem of testing whether a given system of equations
over a fixed finite semigroup S has a solution. For the case where
S is a monoid, we prove that the problem is computable in polynomial
time when S is commutative and is the union of its subgroups
more >>>

TR04-092 | 3rd November 2004
Oded Lachish, Ilan Newman

Testing Periodicity

A string $\alpha\in\Sigma^n$ is called {\it p-periodic},
if for every $i,j \in \{1,\dots,n\}$, such that $i\equiv j \bmod p$,
$\alpha_i = \alpha_{j}$, where $\alpha_i$ is the $i$-th place of $\alpha$.
A string $\alpha\in\Sigma^n$ is said to be $period(\leq g)$,
if there exists $p\in \{1,\dots,g\}$ such that $\alpha$ ... more >>>

TR04-093 | 9th November 2004
Oded Goldreich, Madhu Sudan, Luca Trevisan

From logarithmic advice to single-bit advice

Building on Barak's work (Random'02),
Fortnow and Santhanam (FOCS'04) proved a time hierarchy
for probabilistic machines with one bit of advice.
Their argument is based on an implicit translation technique,
which allow to translate separation results for short (say logarithmic)
advice (as shown by Barak) into separations for ... more >>>

TR04-094 | 10th November 2004
Omer Reingold

Undirected ST-Connectivity in Log-Space

We present a deterministic, log-space algorithm that solves
st-connectivity in undirected graphs. The previous bound on the
space complexity of undirected st-connectivity was
log^{4/3}() obtained by Armoni, Ta-Shma, Wigderson and
Zhou. As undirected st-connectivity is
complete for the class of problems solvable by symmetric,
non-deterministic, log-space computations (the class SL), ... more >>>

TR04-095 | 3rd November 2004
Daniele Micciancio

Generalized compact knapsacks, cyclic lattices, and efficient one-way functions from worst-case complexity assumptions

We investigate the average case complexity of a generalization of the compact knapsack problem to arbitrary rings: given $m$ (random) ring elements a_1,...,a_m in R and a (random) target value b in R, find coefficients x_1,...,x_m in S (where S is an appropriately chosen subset of R) such that a_1*x_1 ... more >>>

TR04-096 | 4th November 2004
Eldar Fischer, Frederic Magniez, Michel de Rougemont

Property and Equivalence Testing on Strings

Revisions: 1

Using a new statistical embedding of words which has similarities with the Parikh mapping, we first construct a tolerant tester for the equality of two words, whose complexity is independent of the string size, where the distance between inputs is measured by the normalized edit distance with moves. As a ... more >>>

TR04-097 | 2nd November 2004
Víctor Dalmau

Malt'sev Constraints made Simple

We give in this paper a different and simpler proof of the tractability of Mal'tsev contraints.

more >>>

TR04-098 | 5th November 2004
Lance Fortnow, Rahul Santhanam, Luca Trevisan

Promise Hierarchies

We show that for any constant a, ZPP/b(n) strictly contains
ZPTIME(n^a)/b(n) for some b(n) = O(log n log log n). Our techniques
are very general and give the same hierarchy for all the common
promise time classes including RTIME, NTIME \cap coNTIME, UTIME,

We show a ... more >>>

TR04-099 | 11th November 2004
Ran Raz

Extractors with Weak Random Seeds

We show how to extract random bits from two or more independent
weak random sources, in cases where only one source is of linear
min-entropy and all other sources are of logarithmic min-entropy.
We also give improved constructions of mergers and condensers.
In all that comes below, $\delta$ is an ... more >>>

TR04-100 | 23rd November 2004
Eric Allender, Michael Bauland, Neil Immerman, Henning Schnoor, Heribert Vollmer

The Complexity of Satisfiability Problems: Refining Schaefer's Theorem

Revisions: 1

Schaefer proved in 1978 that the Boolean constraint satisfaction problem for a given constraint language is either in P or is NP-complete, and identified all tractable cases. Schaefer's dichotomy theorem actually shows that there are at most two constraint satisfaction problems, up to polynomial-time isomorphism (and these isomorphism types are ... more >>>

TR04-101 | 28th September 2004
Miroslav Chlebík, Janka Chlebíková

Crown reductions for the Minimum Weighted Vertex Cover problem

TR04-102 | 20th October 2004
Thomas Holenstein

Key Agreement from Weak Bit Agreement

Assume that Alice and Bob, given an authentic channel, have a protocol where they end up with a bit S_A and S_B, respectively, such that with probability (1+eps)/2 these bits are equal. Further assume that conditioned on the event S_A = S_B no polynomial time bounded algorithm can predict the ... more >>>

TR04-103 | 19th November 2004
Lance Fortnow, Adam Klivans

NP with Small Advice

We prove a new equivalence between the non-uniform and uniform complexity of exponential time. We show that EXP in NP/log if and only if EXP = P^NP|| (polynomial time with non-adaptive queries to SAT). Our equivalence makes use of a recent result due to Shaltiel and Umans showing EXP in ... more >>>

TR04-104 | 19th November 2004
Maria Lopez-Valdes, Mayordomo Elvira

Dimension is compression

Effective fractal dimension was defined by Lutz (2003) in order to quantitatively analyze the structure of complexity classes.
Interesting connections of effective dimension with information theory were also found, in fact the cases of polynomial-space and constructive dimension can be precisely characterized in terms of Kolmogorov complexity, while analogous ... more >>>

TR04-105 | 19th November 2004
Eldar Fischer, Lance Fortnow

Tolerant Versus Intolerant Testing for Boolean Properties

A property tester with high probability accepts inputs satisfying a given property and rejects
inputs that are far from satisfying it. A tolerant property tester, as defined by Parnas, Ron
and Rubinfeld, must also accept inputs that are close enough to satisfying the property. We
construct properties of binary functions ... more >>>

TR04-106 | 19th November 2004
Christian Glaßer, Alan L. Selman, Liyu Zhang

Canonical Disjoint NP-Pairs of Propositional Proof Systems

We prove that every disjoint NP-pair is polynomial-time, many-one equivalent to
the canonical disjoint NP-pair of some propositional proof system. Therefore, the degree structure of the class of disjoint NP-pairs and of all canonical pairs is
identical. Secondly, we show that this degree structure is not superficial: Assuming there exist ... more >>>

TR04-107 | 24th November 2004
Ingo Wegener, Philipp Woelfel

New Results on the Complexity of the Middle Bit of Multiplication

Revisions: 1

It is well known that the hardest bit of integer multiplication is the middle bit, i.e. MUL_{n-1,n}.
This paper contains several new results on its complexity.
First, the size s of randomized read-k branching programs, or, equivalently, its space (log s) is investigated.
A randomized algorithm for MUL_{n-1,n} with k=O(log ... more >>>

TR04-108 | 24th November 2004
Eric Allender, Samir Datta, Sambuddha Roy

Topology inside NC^1

We show that ACC^0 is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC^0. Thus polylogarithmic genus provides no additional computational power in this model.
We consider other generalizations of ... more >>>

TR04-109 | 15th November 2004
Neeraj Kayal, Nitin Saxena

On the Ring Isomorphism & Automorphism Problems

We study the complexity of the isomorphism and automorphism problems for finite rings with unity.

We show that both integer factorization and graph isomorphism reduce to the problem of counting
automorphisms of rings. The problem is shown to be in the complexity class $\AM \cap co\AM$
and hence ... more >>>

TR04-110 | 24th November 2004
Tomoyuki Yamakami, Harumichi Nishimura

An Application of Quantum Finite Automata to Interactive Proof Systems

Quantum finite automata have been studied intensively since
their introduction in late 1990s as a natural model of a
quantum computer with finite-dimensional quantum memory space.
This paper seeks their direct application
to interactive proof systems in which a mighty quantum prover
communicates with a ... more >>>

TR04-111 | 30th November 2004
Piotr Berman, Marek Karpinski, Alexander D. Scott, Alexander D. Scott

Computational Complexity of Some Restricted Instances of 3SAT

We prove results on the computational complexity of instances of 3SAT in which every variable occurs 3 or 4 times.

more >>>

TR04-112 | 26th November 2004
Neil Thapen, Nicola Galesi

Resolution and pebbling games

We define a collection of Prover-Delayer games that characterize certain subsystems of resolution. This allows us to give some natural criteria which guarantee lower bounds on the resolution width of a formula, and to extend these results to formulas of unbounded initial width.

We also use games to give upper ... more >>>

TR04-113 | 19th November 2004
Mårten Trolin

Lattices with Many Cycles Are Dense

We give a method for approximating any $n$-dimensional
lattice with a lattice $\Lambda$ whose factor group
$\mathbb{Z}^n / \Lambda$ has $n-1$ cycles of equal length
with arbitrary precision. We also show that a direct
consequence of this is that the Shortest Vector Problem and the Closest
Vector Problem cannot ... more >>>

TR04-114 | 21st November 2004
Vladimir Trifonov

An O(log n log log n) Space Algorithm for Undirected s,t-Connectivity

We present a deterministic O(log n log log n) space algorithm for
undirected s,t-connectivity. It is based on the deterministic EREW
algorithm of Chong and Lam (SODA 93) and uses the universal
exploration sequences for trees constructed by Kouck\'y (CCC 01).
Our result improves the O(log^{4/3} n) bound of Armoni ... more >>>

TR04-115 | 1st December 2004
Iftach Haitner, Ronen Shaltiel

Statistical Zero-Knowledge Arguments for NP Using Approximable-Preimage-Size One-Way Functions

A statistical zero knowledge argument for NP is a cryptographic primitive that allows a polynomial-time prover to convince another
polynomial-time verifier of the validity of an NP statement. It is guaranteed that even an infinitely powerful verifier does not learn any
additional information but the validity of the claim.

Naor ... more >>>

TR04-116 | 18th November 2004
PERRET ludovic

On the computational complexity of some equivalence problems of polynomial systems of equations over finite fields

We study in this paper the computational complexity of some
equivalence relations on polynomial systems of equations over finite
fields. These problems are analyzed with respect to polynomial-time
many-one reductions (resp. Turing reductions, Levin reductions). In
particular, we show that some of these problems are between ... more >>>

TR04-117 | 1st December 2004
Michael Alekhnovich, Sanjeev Arora, Iannis Tourlakis

Towards strong nonapproximability results in the Lovasz-Schrijver hierarchy

Lovasz and Schrijver described a generic method of tightening the LP and SDP relaxation for any 0-1 optimization problem. These tightened relaxations were the basis of several celebrated approximation algorithms (such as for MAX-CUT, MAX-3SAT, and SPARSEST CUT).

We prove strong nonapproximability results in this model for well-known problems such ... more >>>

TR04-118 | 21st December 2004
Marek Karpinski, Yakov Nekrich

A Note on Traversing Skew Merkle Trees

We consider the problem of traversing skew (unbalanced) Merkle
trees and design an algorithm for traversing a skew Merkle tree
in time O(log n/log t) and space O(log n (t/log t)), for any choice
of parameter t\geq 2.
This algorithm can be of special interest in situations when
more >>>

TR04-119 | 8th December 2004
Uriel Feige, Daniel Reichman

On The Hardness of Approximating Max-Satisfy

Max-Satisfy is the problem of finding an assignment that satisfies
the maximum number of equations in a system of linear equations
over $\mathbb{Q}$. We prove that unless NP$\subseteq $BPP there is no
polynomial time algorithm for the problem achieving an
approximation ratio of $1/n^{1-\epsilon}$, where $n$ is the number
of ... more >>>

TR04-120 | 22nd November 2004
Andris Ambainis, William Gasarch, Aravind Srinivasan, Andrey Utis

Lower bounds on the Deterministic and Quantum Communication Complexity of HAM_n^a

Alice and Bob want to know if two strings of length $n$ are
almost equal. That is, do they differ on at most $a$ bits?
Let $0\le a\le n-1$.
We show that any deterministic protocol, as well as any
error-free quantum protocol ($C^*$ version), for this problem
requires at ... more >>>

TR04-121 | 7th December 2004
Vikraman Arvind, Piyush Kurur, T.C. Vijayaraghavan

Bounded Color Multiplicity Graph Isomorphism is in the #L Hierarchy.

In this paper we study the complexity of Bounded Color
Multiplicity Graph Isomorphism (BCGI): the input is a pair of
vertex-colored graphs such that the number of vertices of a given
color in an input graph is bounded by $b$. We show that BCGI is in the
#L hierarchy ... more >>>

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