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Electronic Colloquium on Computational Complexity

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REPORTS > 2003:
All reports in year 2003:
TR03-001 | 8th January 2003
Vince Grolmusz

Near Quadratic Matrix Multiplication Modulo Composites

Comments: 1

We show how one can use non-prime-power, composite moduli for
computing representations of the product of two $n\times n$ matrices
using only $n^{2+o(1)}$ multiplications.

more >>>

TR03-002 | 13th December 2002
Stefan Szeider

Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable

Revisions: 1

The deficiency of a propositional formula F in CNF with n variables
and m clauses is defined as m-n. It is known that minimal
unsatisfiable formulas (unsatisfiable formulas which become
satisfiable by removing any clause) have positive deficiency.
Recognition of minimal unsatisfiable formulas is NP-hard, and it was
shown recently ... more >>>


TR03-003 | 19th December 2002
Fahiem Bacchus, Shannon Dalmao

DPLL with Caching: A new algorithm for #SAT and Bayesian Inference

Bayesian inference and counting satisfying assignments are important
problems with numerous applications in probabilistic reasoning. In this
paper, we show that plain old DPLL equipped with memoization can solve
both of these problems with time complexity that is at least as
good as all known algorithms. Furthermore, DPLL with memoization
more >>>


TR03-004 | 24th December 2002
Eli Ben-Sasson, Prahladh Harsha

Lower Bounds for Bounded-Depth Frege Proofs via Buss-Pudlack Games

We present a simple proof of the bounded-depth Frege lower bounds of
Pitassi et. al. and Krajicek et. al. for the pigeonhole
principle. Our method uses the interpretation of proofs as two player
games given by Pudlak and Buss. Our lower bound is conceptually
simpler than previous ones, and relies ... more >>>


TR03-005 | 28th December 2002
Scott Aaronson

Quantum Certificate Complexity

Given a Boolean function f, we study two natural generalizations of the certificate complexity C(f): the randomized certificate complexity RC(f) and the quantum certificate complexity QC(f). Using Ambainis' adversary method, we exactly characterize QC(f) as the square root of RC(f). We then use this result to prove the new relation ... more >>>


TR03-006 | 23rd January 2003
Eli Ben-Sasson, Prahladh Harsha, Sofya Raskhodnikova

3CNF Properties are Hard to Test

For a boolean formula \phi on n variables, the associated property
P_\phi is the collection of n-bit strings that satisfy \phi. We prove
that there are 3CNF properties that require a linear number of queries,
even for adaptive tests. This contrasts with 2CNF properties
that are testable with O(\sqrt{n}) ... more >>>


TR03-007 | 15th January 2003
Olivier Dubois, Yacine Boufkhad, Jacques Mandler

Typical random 3-SAT formulae and the satisfiability threshold

$k$-SAT is one of the best known among a wide class of random
constraint satisfaction problems believed to exhibit a threshold
phenomenon where the control parameter is the ratio, number of
constraints to number of variables. There has been a large amount of
work towards estimating ... more >>>


TR03-008 | 11th February 2003
Piotr Berman, Marek Karpinski

Improved Approximation Lower Bounds on Small Occurrence Optimization

We improve a number of approximation lower bounds for
bounded occurrence optimization problems like MAX-2SAT,
E2-LIN-2, Maximum Independent Set and Maximum-3D-Matching.

more >>>

TR03-009 | 3rd February 2003
Markus Bläser, Andreas Jakoby, Maciej Liskiewicz, Bodo Manthey

Private Computation --- $k$-connected versus $1$-connected Networks

Revisions: 1

We study the role of connectivity of communication networks in private
computations under information theoretic settings. It will be shown that
some functions can be computed by private protocols even if the
underlying network is 1-connected but not 2-connected. Then we give a
complete characterisation of non-degenerate functions that can ... more >>>


TR03-010 | 13th February 2003
Sven Baumer, Rainer Schuler

Improving a probabilistic 3-SAT Algorithm by Dynamic Search and Independent Clause Pairs

The satisfiability problem of Boolean Formulae in 3-CNF (3-SAT)
is a well known NP-complete problem and the development of faster
(moderately exponential time) algorithms has received much interest
in recent years. We show that the 3-SAT problem can be solved by a
probabilistic algorithm in expected time O(1,3290^n).
more >>>


TR03-011 | 17th February 2003
Christian Glaßer, Alan L. Selman, Samik Sengupta, Liyu Zhang

Disjoint NP-Pairs

We study the question of whether the class DisNP of
disjoint pairs (A, B) of NP-sets contains a complete pair.
The question relates to the question of whether optimal
proof systems exist, and we relate it to the previously
studied question of whether there exists ... more >>>


TR03-012 | 21st January 2003
Edward Hirsch, Arist Kojevnikov

Several notes on the power of Gomory-Chvatal cuts

We prove that the Cutting Plane proof system based on
Gomory-Chvatal cuts polynomially simulates the
lift-and-project system with integer coefficients
written in unary. The restriction on coefficients can be
omitted when using Krajicek's cut-free Gentzen-style extension
of both systems. We also prove that Tseitin tautologies
have short proofs in ... more >>>


TR03-013 | 7th March 2003
Luca Trevisan

An epsilon-Biased Generator in NC0

Comments: 1

Cryan and Miltersen recently considered the question
of whether there can be a pseudorandom generator in
NC0, that is, a pseudorandom generator such that every
bit of the output depends on a constant number k of bits
of the seed. They show that for k=3 there ... more >>>


TR03-014 | 28th February 2003
Avrim Blum, Ke Yang

On Statistical Query Sampling and NMR Quantum Computing

We introduce a ``Statistical Query Sampling'' model, in which
the goal of an algorithm is to produce an element in a hidden set
$S\subseteq\bit^n$ with reasonable probability. The algorithm
gains information about $S$ through oracle calls (statistical
queries), where the algorithm submits a query function $g(\cdot)$
and receives ... more >>>


TR03-015 | 20th March 2003
Yael Tauman

On the (In)security of the Fiat-Shamir Paradigm

In 1986, Fiat and Shamir suggested a general method for transforming secure 3-round public-coin identification schemes into digital signature schemes. The significant contribution of this method is a means for designing efficient digital signatures, while hopefully achieving security against chosen message attacks. All other known constructions which achieve such security ... more >>>


TR03-016 | 15th January 2003
Dimitrios Koukopoulos, Marios Mavronicolas, Paul Spirakis

FIFO is Unstable at Arbitrarily Low Rates

Revisions: 1

In this work, we study the stability of the {\sf FIFO} ({\sf
First-In-First-Out}) protocol in the context of Adversarial
Queueing Theory. As an important intermediate step, we consider
{\em dynamic capacities}, where each network link capacity may
arbitrarily take on values in the two-valued set of integers
$\{1,C\}$ for $C>1$ ... more >>>


TR03-017 | 27th March 2003
Peter Bro Miltersen, Jaikumar Radhakrishnan, Ingo Wegener

On Converting CNF to DNF

The best-known representations of boolean functions f are those of disjunctions of terms (DNFs) and as conjuctions of clauses (CNFs). It is convenient to define the DNF size of f as the minimal number of terms in a DNF representing f and the CNF size as the minimal number of ... more >>>


TR03-018 | 28th March 2003
Matthias Galota, Heribert Vollmer

Functions Computable in Polynomial Space

We show that the class of integer-valued functions computable by
polynomial-space Turing machines is exactly the class of functions f
for which there is a nondeterministic polynomial-time Turing
machine with a certain order on its paths that on input x outputs a 3x3
matrix with entries from {-1,0,1} on each ... more >>>


TR03-019 | 3rd April 2003
Eli Ben-Sasson, Oded Goldreich, Madhu Sudan

Bounds on 2-Query Codeword Testing.

Revisions: 1


We present upper bounds on the size of codes that are locally
testable by querying only two input symbols. For linear codes, we
show that any $2$-locally testable code with minimal distance
$\delta n$ over a finite field $F$ cannot have more than
$|F|^{3/\delta}$ codewords. This result holds even ... more >>>


TR03-020 | 27th March 2003
Elad Hazan, Shmuel Safra, Oded Schwartz

On the Hardness of Approximating k-Dimensional Matching

We study bounded degree graph problems, mainly the problem of
k-Dimensional Matching \emph{(k-DM)}, namely, the problem of
finding a maximal matching in a k-partite k-uniform balanced
hyper-graph. We prove that k-DM cannot be efficiently approximated
to within a factor of $ O(\frac{k}{ \ln k}) $ unless $P = NP$.
This ... more >>>


TR03-021 | 4th April 2003
Mikhail Vyalyi

QMA=PP implies that PP contains PH

We consider possible equality QMA=PP and give an argument
against it. Namely, this equality implies that PP contains PH. The argument is based on the strong form of Toda's theorem and
the strengthening of the proof for inclusion $QMA\subseteq PP$ due to Kitaev and Watrous.

more >>>

TR03-022 | 11th April 2003
Piotr Berman, Marek Karpinski, Alexander D. Scott

Approximation Hardness and Satisfiability of Bounded Occurrence Instances of SAT

We study approximation hardness and satisfiability of bounded
occurrence uniform instances of SAT. Among other things, we prove
the inapproximability for SAT instances in which every clause has
exactly 3 literals and each variable occurs exactly 4 times,
and display an explicit ... more >>>


TR03-023 | 12th February 2003
Anna Palbom

On Spanning Cacti and Asymmetric TSP

In an attempt to generalize Christofides algorithm for metric TSP to the asymmetric TSP with triangle inequality we have studied various properties of directed spanning cacti. In this paper we first observe that finding the TSP in a directed, weighted complete graph with triangle inequality is polynomial time equivalent to ... more >>>


TR03-024 | 25th February 2003
Till Tantau

Weak Cardinality Theorems for First-Order Logic

Kummer's cardinality theorem states that a language is recursive
if a Turing machine can exclude for any n words one of the
n + 1 possibilities for the number of words in the language. It
is known that this theorem does not hold for polynomial-time
computations, but there ... more >>>


TR03-025 | 14th April 2003
Kristoffer Arnsfelt Hansen

Constant width planar computation characterizes ACC0

We obtain a characterization of ACC0 in terms of a natural class of
constant width circuits, namely in terms of constant width polynomial
size planar circuits. This is shown via a characterization of the
class of acyclic digraphs which can be embedded on a cylinder surface
in such a way ... more >>>


TR03-026 | 20th February 2003
Janka Chlebíková, Miroslav Chlebík

Inapproximability results for bounded variants of optimization problems

We study small degree graph problems such as Maximum Independent Set
and Minimum Node Cover and improve approximation lower bounds for
them and for a number of related problems, like Max-B-Set Packing,
Min-B-Set Cover, Max-Matching in B-uniform 2-regular hypergraphs.
For example, we prove NP-hardness factor of 95/94
more >>>


TR03-027 | 21st April 2003
Christian Glaßer, Alan L. Selman, Samik Sengupta

Reductions between Disjoint NP-Pairs

We prove that all of the following assertions are equivalent:
There is a many-one complete disjoint NP-pair;
there is a strongly many-one complete disjoint NP-pair;
there is a Turing complete disjoint NP-pair such that all reductions
are smart reductions;
there is a complete disjoint NP-pair for one-to-one, invertible ... more >>>


TR03-028 | 28th February 2003
Olivier Powell

PSPACE contains almost complete problems

An almost complete set A for a complexity class C is a language of C which is not complete, but that has the property that ``many'' languages of C reduce to A, where the term ``many'' is used in reference to Lutz's resource bounded measure (rbm). The question of the ... more >>>


TR03-029 | 1st April 2003
Philippe Moser

BPP has effective dimension at most 1/2 unless BPP=EXP

We prove that BPP has Lutz's p-dimension at most 1/2 unless BPP equals EXP.
Next we show that BPP has Lutz's p-dimension zero unless BPP equals EXP
on infinitely many input lengths.
We also prove that BPP has measure zero in the smaller complexity
class ... more >>>


TR03-030 | 27th February 2003
Amin Coja-Oghlan, Andreas Goerdt, André Lanka, Frank Schädlich

Certifying Unsatisfiability of Random 2k-SAT Formulas using Approximation Techniques

Abstract. It is known that random k-SAT formulas with at least
(2^k*ln2)*n random clauses are unsatisfiable with high probability. This
result is simply obtained by bounding the expected number of satisfy-
ing assignments of a random k-SAT instance by an expression tending
to 0 when n, the number of variables ... more >>>


TR03-031 | 8th April 2003
Birgit Schelm

Average-Case Complexity Theory of Approximation Problems

Both average-case complexity and the study of the approximability properties of NP-optimization problems are well established and active fields of research. By applying the notion of average-case complexity to approximation problems we provide a formal framework that allows the classification of NP-optimization problems according to their average-case approximability. Thus, known ... more >>>


TR03-032 | 16th April 2003
Andreas Björklund, Thore Husfeldt, Sanjeev Khanna

Approximating Longest Directed Path

We investigate the hardness of approximating the longest path and
the longest cycle in directed graphs on $n$ vertices. We show that
neither of these two problems can be polynomial time approximated
within $n^{1-\epsilon}$ for any $\epsilon>0$ unless
$\text{P}=\text{NP}$. In particular, the result holds for
more >>>


TR03-033 | 6th May 2003
Meir Feder, Dana Ron, Ami Tavory

Bounds on Linear Codes for Network Multicast

Comments: 1

Traditionally, communication networks are composed of
routing nodes, which relay and duplicate data. Work in
recent years has shown that for the case of multicast, an
improvement in both rate and code-construction complexity can be
gained by replacing these routing nodes by linear coding
nodes. These nodes transmit linear combinations ... more >>>


TR03-034 | 17th March 2003
Arnold Beckmann

Height restricted constant depth LK

Height restricted constant depth LK is a natural restriction of
Gentzen's propositional proof system LK. A sequence of LK-formulas
has polylogarithmic-height restricted depth-d-LK proofs iff the
n-th formula in the sequence possesses LK-proofs where cut-formulas
are of depth d+1 with small bottom fanin
and of ... more >>>


TR03-035 | 21st May 2003
Eran Halperin, Guy Kortsarz, Robert Krauthgamer

Tight lower bounds for the asymmetric k-center problem

In the {\sc $k$-center} problem, the input is a bound $k$
and $n$ points with the distance between every two of them,
such that the distances obey the triangle inequality.
The goal is to choose a set of $k$ points to serve as centers,
so that the maximum distance ... more >>>


TR03-036 | 27th April 2003
Bruce Edward Litow

Polynomial equation elimination via Tarski Algebra

The elimination
problem is classical:
implicitly express one of the variables occurring in a finite
system of polynomial equations as an algebraic function of a
designated subset of the remaining variables. Solutions to this
problem by resultants, or more comprehensively by
use of Gr\"{o}bner basis methods are available. In this ... more >>>


TR03-037 | 30th April 2003
Ziv Bar-Yossef

Sampling Lower Bounds via Information Theory

We present a novel technique, based on the Jensen-Shannon divergence
from information theory, to prove lower bounds on the query complexity
of sampling algorithms that approximate functions over arbitrary
domain and range. Unlike previous methods, our technique does not
use a reduction from a binary decision problem, but rather ... more >>>


TR03-038 | 15th May 2003
Julia Chuzhoy, Sudipto Guha, Sanjeev Khanna, Seffi Naor

Asymmetric k-center is log^*n-hard to Approximate

We show that the asymmetric $k$-center problem is
$\Omega(\log^* n)$-hard to approximate unless
${\rm NP} \subseteq {\rm DTIME}(n^{poly(\log \log n)})$.
Since an $O(\log^* n)$-approximation algorithm is known
for this problem, this essentially resolves the approximability
of this problem. This is the first natural problem
whose approximability threshold does not polynomially ... more >>>


TR03-039 | 19th May 2003
Judy Goldsmith, Robert H. Sloan, Balázs Szörényi, György Turán

Theory Revision with Queries: Horn, Read-once, and Parity Formulas

A theory, in this context, is a Boolean formula; it is
used to classify instances, or truth assignments. Theories
can model real-world phenomena, and can do so more or less
correctly.
The theory revision, or concept revision, problem is to
correct a given, roughly correct concept.
This problem is ... more >>>


TR03-040 | 3rd June 2003
Philippe Moser

RP is Small in SUBEXP else ZPP equals PSPACE and NP equals EXP

We use recent results on the hardness of resource-bounded
Kolmogorov random strings, to prove that RP is small in SUBEXP
else ZPP=PSPACE and NP=EXP.
We also prove that if NP is not small in SUBEXP, then
NP=AM, improving a former result which held for the measure ... more >>>


TR03-041 | 29th May 2003
Albert Atserias, Maria Luisa Bonet, Jordi Levy

On Chvatal Rank and Cutting Planes Proofs

We study the Chv\'atal rank of polytopes as a complexity measure of
unsatisfiable sets of clauses. Our first result establishes a
connection between the Chv\'atal rank and the minimum refutation
length in the cutting planes proof system. The result implies that
length lower bounds for cutting planes, or even for ... more >>>


TR03-042 | 15th May 2003
Luca Trevisan

List Decoding Using the XOR Lemma

We show that Yao's XOR Lemma, and its essentially equivalent
rephrasing as a Direct Product Lemma, can be
re-interpreted as a way of obtaining error-correcting
codes with good list-decoding algorithms from error-correcting
codes having weak unique-decoding algorithms. To get codes
with good rate and efficient list decoding algorithms
one needs ... more >>>


TR03-043 | 14th May 2003
Elchanan Mossel, Amir Shpilka, Luca Trevisan

On epsilon-Biased Generators in NC0

Cryan and Miltersen recently considered the question
of whether there can be a pseudorandom generator in
NC0, that is, a pseudorandom generator such that every
bit of the output depends on a constant number k of bits
of the seed. They show that for k=3 there is always a
distinguisher; ... more >>>


TR03-044 | 12th May 2003
Juan Luis Esteban, Jacobo Toran

A Combinatorial Characterization of Treelike Resolution Space

We show that the Player-Adversary game from a paper
by Pudlak and Impagliazzo played over
CNF propositional formulas gives
an exact characterization of the space needed
in treelike resolution refutations. This
characterization is purely combinatorial
and independent of the notion of resolution.
We use this characterization to give ... more >>>


TR03-045 | 8th June 2003
Oded Goldreich, Asaf Nussboim

On the Implementation of Huge Random Objects

Revisions: 1


We initiate a general study of pseudo-random implementations
of huge random objects, and apply it to a few areas
in which random objects occur naturally.
For example, a random object being considered may be
a random connected graph, a random bounded-degree graph,
or a random error-correcting code with good ... more >>>


TR03-046 | 11th June 2003
Philippe Moser

Locally Computed Baire's Categories on Small Complexity Classes

We strengthen an earlier notion of
resource-bounded Baire's categories, and define
resource bounded Baire's categories on small complexity classes such as P, QP, SUBEXP
and on probabilistic complexity classes such as BPP.
We give an alternative characterization of meager sets via resource-bounded
Banach Mazur games.
We show that the class ... more >>>


TR03-047 | 22nd June 2003
Nayantara Bhatnagar, Parikshit Gopalan, Richard J. Lipton

Symmetric Polynomials over Z_m and Simultaneous Communication Protocols

We study the problem of representing symmetric Boolean functions as symmetric polynomials over Z_m. We show an equivalence between such
representations and simultaneous communication protocols. Computing a function with a polynomial of degree d modulo m=pq is equivalent to a two player protocol where one player is given the first ... more >>>


TR03-048 | 24th June 2003
Stefan Droste, Thomas Jansen, Ingo Wegener

Upper and Lower Bounds for Randomized Search Heuristics in Black-Box Optimization

Randomized search heuristics like local search, simulated annealing or all kinds of evolutionary algorithms have many applications. However, for most problems the best worst-case expected run times are achieved by more problem-specific algorithms. This raises the question about the limits of general randomized search heuristics.

Here a framework called black-box ... more >>>


TR03-049 | 25th June 2003
Piotr Berman, Marek Karpinski, Alexander D. Scott

Approximation Hardness of Short Symmetric Instances of MAX-3SAT

We prove approximation hardness of short symmetric instances
of MAX-3SAT in which each literal occurs exactly twice, and
each clause is exactly of size 3. We display also an explicit
approximation lower bound for that problem. The bound two on
the number ... more >>>


TR03-050 | 16th June 2003
Daniel Král

Locally satisfiable formulas

A CNF formula is k-satisfiable if each k clauses of it can be satisfied
simultaneously. Let \pi_k be the largest real number such that for each
k-satisfiable formula with variables x_i, there are probabilities p_i
with the following property: If each variable x_i is chosen randomly and
independently to be ... more >>>


TR03-051 | 20th June 2003
Tsuyoshi Morioka

The Relative Complexity of Local Search Heuristics and the Iteration Principle

Johnson, Papadimitriou and Yannakakis introduce the class $\PLS$
consisting of optimization problems for which efficient local-
search heuristics exist. We formulate a type-2 problem $\iter$
that characterizes $\PLS$ in style of Beame et al., and prove
a criterion for type-2 problems to be nonreducible to $\iter$.
As a corollary, ... more >>>


TR03-052 | 13th May 2003
Stanislav Busygin, Dmitrii V. Pasechnik

On ~chi(G)-alpha(G)>0 gap recognition and alpha(G)-upper bounds

We show that for a graph G it is NP-hard to decide whether its independence number alpha(G) equals its clique partition number ~chi(G) even when some minimum clique partition of G is given. This implies that any alpha(G)-upper bound provably better than ~chi(G) is NP-hard to compute.

To establish this ... more >>>


TR03-053 | 8th July 2003
Kazuo Iwama, Suguru Tamaki

Improved Upper Bounds for 3-SAT

This paper presents a new upper bound for the
$k$-satisfiability problem. For small $k$'s, especially for $k=3$,
there have been a lot of algorithms which run significantly faster
than the trivial $2^n$ bound. The following list summarizes those
algorithms where a constant $c$ means that the algorithm runs in time
more >>>


TR03-054 | 2nd July 2003
Daniel Rolf

3-SAT in RTIME(O(1.32793^n)) - Improving Randomized Local Search by Initializing Strings of 3-Clauses

This paper establishes a randomized algorithm that finds a satisfying assignment for a satisfiable formula $F$ in 3-CNF in $O(1.32793^n)$ expected running time. The algorithms is based on the analysis of so-called strings, which are sequences of 3-clauses where non-succeeding clauses do not share a variable and succeeding clauses share ... more >>>


TR03-055 | 20th July 2003
Jan Krajicek

Implicit proofs


We describe a general method how to construct from
a propositional proof system P a possibly much stronger
proof system iP. The system iP operates with
exponentially long P-proofs described ``implicitly''
by polynomial size circuits.

As an example we prove that proof system iEF, implicit EF,
corresponds to bounded ... more >>>


TR03-056 | 29th July 2003
Piotr Berman, Marek Karpinski

Approximability of Hypergraph Minimum Bisection

We prove that the problems of minimum bisection on k-uniform
hypergraphs are almost exactly as hard to approximate,
up to the factor k/3, as the problem of minimum bisection
on graphs. On a positive side, our argument gives also the
first approximation ... more >>>


TR03-057 | 21st July 2003
Scott Aaronson

Lower Bounds for Local Search by Quantum Arguments

The problem of finding a local minimum of a black-box function is central
for understanding local search as well as quantum adiabatic algorithms.
For functions on the Boolean hypercube {0,1}^n, we show a lower bound of
Omega(2^{n/4}/n) on the number of queries needed by a quantum computer to
solve this ... more >>>


TR03-058 | 22nd July 2003
Vince Grolmusz

Defying Dimensions Modulo 6

Revisions: 2

We show that a certain representation of the matrix-product can be computed with $n^{o(1)}$ multiplications. We also show, that similar representations of matrices can be compressed enormously with the help of simple linear transforms.

more >>>

TR03-059 | 18th May 2003
Harumichi Nishimura, Tomoyuki Yamakami

Polynomial time quantum computation with advice

Revisions: 2

Advice is supplementary information that enhances the computational
power of an underlying computation. This paper focuses on advice that
is given in the form of a pure quantum state. The notion of advised
quantum computation has a direct connection to non-uniform quantum
circuits and tally languages. The paper examines the ... more >>>


TR03-060 | 7th September 2003
Danny Harnik, Moni Naor, Omer Reingold, Alon Rosen

Completeness in Two-Party Secure Computation - A Computational View

A Secure Function Evaluation (SFE) of a two-variable function f(.,.) is a protocol that allows two parties with inputs x and y to evaluate
f(x,y) in a manner where neither party learns ``more than is necessary". A rich body of work deals with the study of completeness for secure ... more >>>


TR03-061 | 29th August 2003
Jan Kára, Daniel Král

Free Binary Decision Diagrams for Computation of EAR_n

Free binary decision diagrams (FBDDs) are graph-based data structures representing Boolean functions with a constraint (additional to binary decision diagrams) that each variable is tested at most once during the computation. The function EAR_n is the following Boolean function defined
for n x n Boolean matrices: EAR_n(M)=1 iff the matrix ... more >>>


TR03-062 | 10th July 2003
Andrei Krokhin, Peter Jonsson

Recognizing Frozen Variables in Constraint Satisfaction Problems

In constraint satisfaction problems over finite domains, some variables
can be frozen, that is, they take the same value in all possible solutions. We study the complexity of the problem of recognizing frozen variables with restricted sets of constraint relations allowed in the
instances. We show that the complexity of ... more >>>


TR03-063 | 2nd July 2003
John Hitchcock

The Size of SPP

Derandomization techniques are used to show that at least one of the
following holds regarding the size of the counting complexity class
SPP.
1. SPP has p-measure 0.
2. PH is contained in SPP.
In other words, SPP is small by being a negligible subset of
exponential time or large ... more >>>


TR03-064 | 23rd June 2003
Vikraman Arvind, Piyush Kurur

Upper Bounds on the Complexity of some Galois Theory Problems

Given a polynomial f(X) with rational coefficients as input
we study the problem of (a) finding the order of the Galois group of
f(X), and (b) determining the Galois group of f(X) by finding a small
generator set. Assuming the generalized Riemann hypothesis, we prove
the following complexity bounds:

1. ... more >>>


TR03-065 | 19th June 2003
Wee, Hoeteck

Compressibility Lower Bounds in Oracle Settings

A source is compressible if we can efficiently compute short
descriptions of strings in the support and efficiently
recover the strings from the descriptions. In this paper, we
present a technique for proving lower bounds on
compressibility in an oracle setting, which yields the
following results:

- We ... more >>>


TR03-066 | 2nd September 2003
Daniele Micciancio

Almost perfect lattices, the covering radius problem, and applications to Ajtai's connection factor

Lattices have received considerable attention as a potential source of computational hardness to be used in cryptography, after a breakthrough result of Ajtai (STOC 1996) connecting the average-case and worst-case complexity of various lattice problems. The purpose of this paper is twofold. On the expository side, we present a rigorous ... more >>>


TR03-067 | 14th August 2003
Ran Raz

Multi-Linear Formulas for Permanent and Determinant are of Super-Polynomial Size

An arithmetic formula is multi-linear if the polynomial computed
by each of its sub-formulas is multi-linear. We prove that any
multi-linear arithmetic formula for the permanent or the
determinant of an $n \times n$ matrix is of size super-polynomial
in $n$.

more >>>

TR03-068 | 30th July 2003
Matthias Homeister

Lower Bounds for the Sum of Graph--driven Read--Once Parity Branching Programs

We prove the first lower bound for restricted read-once parity branching
programs with unlimited parity nondeterminism where for each input the
variables may be tested according to several orderings.

Proving a superpolynomial lower bound for read-once parity branching
programs is still a challenging open problem. The following variant ... more >>>


TR03-069 | 13th August 2003
Elmar Böhler, Christian Glaßer, Daniel Meister

Small Bounded-Error Computations and Completeness

SBP is a probabilistic promise class located
between MA and AM \cap BPPpath. The first
part of the paper studies the question of whether
SBP has many-one complete sets. We relate
this question to the existence of uniform
enumerations. We construct an oracle relative to
which SBP and AM do ... more >>>


TR03-070 | 19th August 2003
Amit Chakrabarti, Oded Regev

An Optimal Randomised Cell Probe Lower Bound for Approximate Nearest Neighbour Searching

We consider the approximate nearest neighbour search problem on the
Hamming Cube $\b^d$. We show that a randomised cell probe algorithm that
uses polynomial storage and word size $d^{O(1)}$ requires a worst case
query time of $\Omega(\log\log d/\log\log\log d)$. The approximation
factor may be as loose as $2^{\log^{1-\eta}d}$ for any ... more >>>


TR03-071 | 18th August 2003
Markus Bläser, Andreas Jakoby, Maciej Liskiewicz, Bodo Manthey

Privacy in Non-Private Environments

Revisions: 1

We study private computations in information-theoretical settings on
networks that are not 2-connected. Non-2-connected networks are
``non-private'' in the sense that most functions cannot privately be
computed on such networks. We relax the notion of privacy by
introducing lossy private protocols, which generalize private
protocols. We measure the information each ... more >>>


TR03-072 | 15th September 2003
Evgeny Dantsin, Edward Hirsch, Alexander Wolpert

Algorithms for SAT based on search in Hamming balls

We present a simple randomized algorithm for SAT and prove an upper
bound on its running time. Given a Boolean formula F in conjunctive
normal form, the algorithm finds a satisfying assignment for F
(if any) by repeating the following: Choose an assignment A at
random and ... more >>>


TR03-073 | 11th June 2003
Amin Coja-Oghlan

The Lovasz number of random graph

We study the Lovasz number theta along with two further SDP relaxations $\thetI$, $\thetII$
of the independence number and the corresponding relaxations of the
chromatic number on random graphs G(n,p). We prove that \theta is
concentrated about its mean, and that the relaxations of the chromatic
number in the case ... more >>>


TR03-074 | 24th June 2003
Vince Grolmusz

Sixtors and Mod 6 Computations

We consider the following phenomenon: with just one multiplication
we can compute (3u+2v)(3x+2y)= 3ux+4vy mod 6, while computing the same polynomial modulo 5 needs 2 multiplications. We generalize this observation and we define some vectors, called sixtors, with remarkable zero-divisor properties. Using sixtors, we also generalize our earlier result ... more >>>


TR03-075 | 7th September 2003
Agostino Capponi

A tutorial on the Deterministic two-party Communication Complexity

Communication complexity is concerned with the question: how much information do the participants of a communication system need to exchange in order to perform certain tasks? The minimum number of bits that must be communicated is the deterministic communication complexity of $f$. This complexity measure was introduced by Yao \cite{1} ... more >>>


TR03-076 | 8th September 2003
Michael Langberg

Testing the independence number of hypergraphs

A $k$-uniform hypergraph $G$ of size $n$ is said to be $\varepsilon$-far from having an independent set of size $\rho n$ if one must remove at least $\varepsilon n^k$ edges of $G$ in order for the remaining hypergraph to have an independent set of size $\rho n$. In this work, ... more >>>


TR03-077 | 4th September 2003
Till Tantau

Logspace Optimisation Problems and their Approximation Properties

This paper introduces logspace optimisation problems as
analogues of the well-studied polynomial-time optimisation
problems. Similarly to them, logspace
optimisation problems can have vastly different approximation
properties, even though the underlying existence and budget problems
have the same computational complexity. Numerous natural problems
are presented that exhibit such a varying ... more >>>


TR03-078 | 23rd October 2003
Fan Chung, Ron Graham, Jia Mao, Andrew Chi-Chih Yao

Finding Favorites

Comments: 2

We investigate a new type of information-theoretic identification problem, suggested to us by Alan Taylor. In this problem we are given a set of items, more than half of which share a common ``good" value. The other items have various other values which are called ``bad". The only method we ... more >>>


TR03-079 | 8th November 2003
Scott Aaronson

Multilinear Formulas and Skepticism of Quantum Computing

Several researchers, including Leonid Levin, Gerard 't Hooft, and
Stephen Wolfram, have argued that quantum mechanics will break down
before the factoring of large numbers becomes possible. If this is
true, then there should be a natural "Sure/Shor separator" -- that is,
a set of quantum ... more >>>


TR03-080 | 12th November 2003
Venkatesan Guruswami

Better Extractors for Better Codes?

We present an explicit construction of codes that can be list decoded
from a fraction $(1-\eps)$ of errors in sub-exponential time and which
have rate $\eps/\log^{O(1)}(1/\eps)$. This comes close to the optimal
rate of $\Omega(\eps)$, and is the first sub-exponential complexity
construction to beat the rate of $O(\eps^2)$ achieved by ... more >>>


TR03-081 | 10th October 2003
Valentin E. Brimkov, Bruno Codenotti, Valentino Crespi, Reneta Barneva, Mauro Leoncini

Computation of the Lov\'asz Theta Function for Circulant Graphs

The Lov\'asz theta function $\theta(G)$ of a graph $G$ has attracted a lot of attention for its connection with diverse issues, such as communicating without errors and computing large cliques in graphs. Indeed this function enjoys the remarkable property of being computable in polynomial time, despite being sandwitched between clique ... more >>>


TR03-082 | 22nd November 2003
Andris Ambainis, Ke Yang

Towards the Classical Communication Complexity of Entanglement Distillation Protocols with Incomplete Information

Entanglement is an essential resource for quantum communication and quantum computation, similar to shared random bits in the classical world. Entanglement distillation extracts nearly-perfect entanglement from imperfect entangled state. The classical communication complexity of these protocols is the minimal amount of classical information that needs to be exchanged for the ... more >>>


TR03-083 | 21st November 2003
Jan Arpe, Andreas Jakoby, Maciej Liskiewicz

One-Way Communication Complexity of Symmetric Boolean Functions

We study deterministic one-way communication complexity
of functions with Hankel communication matrices.
Some structural properties of such matrices are established
and applied to the one-way two-party communication complexity
of symmetric Boolean functions.
It is shown that the number of required communication bits
does not depend on ... more >>>


TR03-084 | 27th November 2003
Joshua Buresh-Oppenheim, Tsuyoshi Morioka

Relativized NP Search Problems and Propositional Proof Systems

We consider Total Functional $\NP$ ($\TFNP$) search problems. Such problems are based on combinatorial principles that guarantee, through locally checkable conditions, that a solution to the problem exists in an exponentially-large domain, and have the property that any solution has a polynomial-size witness that can be verified in polynomial time. ... more >>>


TR03-085 | 28th November 2003
Ke Yang

On the (Im)possibility of Non-interactive Correlation Distillation

We study the problem of non-interactive correlation distillation
(NICD). Suppose Alice and Bob each has a string, denoted by
$A=a_0a_1\cdots a_{n-1}$ and $B=b_0b_1\cdots b_{n-1}$,
respectively. Furthermore, for every $k=0,1,...,n-1$, $(a_k,b_k)$ is
independently drawn from a distribution $\noise$, known as the ``noise
mode''. Alice and Bob wish to ``distill'' the correlation
more >>>


TR03-086 | 1st December 2003
Amos Beimel, Tal Malkin

A Quantitative Approach to Reductions in Secure Computation

Secure computation is one of the most fundamental cryptographic tasks.
It is known that all functions can be computed securely in the
information theoretic setting, given access to a black box for some
complete function such as AND. However, without such a black box, not
all functions can be securely ... more >>>


TR03-087 | 10th December 2003
Richard Beigel, Lance Fortnow, William Gasarch

A Nearly Tight Bound for Private Information Retrieval Protocols

Comments: 1

We show that any 1-round 2-server Private Information
Retrieval Protocol where the answers are 1-bit long must ask questions
that are at least $n-2$ bits long, which is nearly equal to the known
$n-1$ upper bound. This improves upon the approximately $0.25n$ lower
bound of Kerenidis and de Wolf while ... more >>>




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