Electronic Colloquium on Computational Complexity
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REPORTS > 2005:
All reports in year 2005:
TR05-001 | 1st January 2005
Mario Szegedy

Near optimality of the priority sampling procedure

Based on experimental results N. Duffield, C. Lund and M. Thorup \cite{dlt2} conjectured
that the variance of their highly successful priority sampling procedure
is not larger than the variance of the threshold sampling procedure with sample size one smaller.
The conjecture's significance is that the latter procedure is provably optimal ... more >>>

TR05-002 | 6th January 2005
Magnus Bordewich, Martin Dyer, Marek Karpinski

Path Coupling Using Stopping Times and Counting Independent Sets and Colourings in Hypergraphs

We give a new method for analysing the mixing time of a Markov chain using
path coupling with stopping times. We apply this approach to two hypergraph
problems. We show that the Glauber dynamics for independent sets in a
hypergraph mixes rapidly as long as the maximum degree $\Delta$ of ... more >>>

TR05-003 | 23rd December 2004
Scott Aaronson

Quantum Computing, Postselection, and Probabilistic Polynomial-Time

I study the class of problems efficiently solvable by a quantum computer, given the ability to "postselect" on the outcomes of measurements. I prove that this class coincides with a classical complexity class called PP, or Probabilistic Polynomial-Time. Using this result, I show that several simple changes to the axioms ... more >>>

TR05-004 | 3rd January 2005
Leslie G. Valiant

Memorization and Association on a Realistic Neural Model

A central open question of computational neuroscience is to identify the data structures and algorithms that are used in mammalian cortex to support successive acts of the basic cognitive tasks of memorization and association. This paper addresses the simultaneous challenges of realizing these two distinct tasks with the same data ... more >>>

TR05-005 | 20th December 2004
Tomas Feder

Constraint Satisfaction on Finite Groups with Near Subgroups

Constraint satisfaction on finite groups, with subgroups and their cosets
described by generators, has a polynomial time algorithm. For any given
group, a single additional constraint type that is not a coset of a near
subgroup makes the problem NP-complete. We consider constraint satisfaction on
groups with subgroups, near subgroups, ... more >>>

TR05-006 | 28th December 2004
Edward Hirsch, Sergey I. Nikolenko

Simulating Cutting Plane proofs with restricted degree of falsity by Resolution

Comments: 1

Goerdt (1991) considered a weakened version of the Cutting Plane proof system with a restriction on the degree of falsity of intermediate inequalities. (The degree of falsity of an inequality written in the form $\sum a_ix_i+\sum b_i(1-x_i)\ge c,\ a_i,b_i\ge0$ is its constant term $c$.) He proved a superpolynomial lower bound ... more >>>

TR05-007 | 15th December 2004
Vadim Lyubashevsky

On Random High Density Subset Sums

In the Subset Sum problem, we are given n integers a_1,...,a_n
and a target number t, and are asked to find the subset of the
a_i's such that the sum is t. A version of the subset sum
problem is the Random Modular Subset Sum problem. In this version,
the ... more >>>

TR05-008 | 11th December 2004
Neeraj Kayal

Recognizing permutation functions in polynomial time.

Let $\mathbb{F}_q$ be a finite field and $f(x) \in \mathbb{F}_q(x)$ be a rational function over $\mathbb{F}_q$.
The decision problem {\bf PermFunction} consists of deciding whether $f(x)$ induces a permutation on
the elements of $\mathbb{F}_q$. That is, we want to decide whether the corresponding map
$f : \mathbb{F}_q ... more >>>

TR05-009 | 14th December 2004
David P. Woodruff, Sergey Yekhanin

A Geometric Approach to Information-Theoretic Private Information Retrieval

A t-private private information retrieval (PIR) scheme allows a user to retrieve the i-th bit of an n-bit string x replicated among k servers, while any coalition of up to t servers learns no information about i. We present a new geometric approach to PIR, and obtain (1) A t-private ... more >>>

TR05-010 | 8th December 2004
Olivier Powell

Almost Completeness in Small Complexity Classes

We constructively prove the existence of almost complete problems under logspace manyone reduction for some small complexity classes by exhibiting a parametrizable construction which yields, when appropriately setting the parameters, an almost complete problem for PSPACE, the class of space efficiently decidable problems, and for SUBEXP, the class of problems ... more >>>

TR05-011 | 21st December 2004
Christian Glaßer, Mitsunori Ogihara, A. Pavan, Alan L. Selman, Liyu Zhang

Autoreducibility, Mitoticity, and Immunity

We show the following results regarding complete sets:

NP-complete sets and PSPACE-complete sets are many-one

Complete sets of any level of PH, MODPH, or
the Boolean hierarchy over NP are many-one autoreducible.

EXP-complete sets are many-one mitotic.

NEXP-complete sets are weakly many-one mitotic.

PSPACE-complete sets are weakly Turing-mitotic.

... more >>>

TR05-012 | 17th January 2005
Luca Trevisan, Salil Vadhan, David Zuckerman

Compression of Samplable Sources

We study the compression of polynomially samplable sources. In particular, we give efficient prefix-free compression and decompression algorithms for three classes of such sources (whose support is a subset of {0,1}^n).

1. We show how to compress sources X samplable by logspace machines to expected length H(X)+O(1).

Our next ... more >>>

TR05-013 | 22nd December 2004
Bin Fu

Theory and Application of Width Bounded Geometric Separator

We introduce the notion of width bounded geometric separator,
develop the techniques for its existence as well as algorithm, and
apply it to obtain a $2^{O(\sqrt{n})}$ time exact algorithm for the
disk covering problem, which seeks to determine the minimal number
of fixed size disks to cover $n$ points on ... more >>>

TR05-014 | 30th January 2005
Oded Goldreich

Short Locally Testable Codes and Proofs (Survey)

We survey known results regarding locally testable codes
and locally testable proofs (known as PCPs),
with emphasis on the length of these constructs.
Locally testability refers to approximately testing
large objects based on a very small number of probes,
each retrieving a single bit in the ... more >>>

TR05-015 | 27th January 2005
Andrej Bogdanov, Luca Trevisan

On Worst-Case to Average-Case Reductions for NP Problems

We show that if an NP-complete problem has a non-adaptive
self-corrector with respect to a samplable distribution then
coNP is contained in NP/poly and the polynomial
hierarchy collapses to the third level. Feigenbaum and
Fortnow (SICOMP 22:994-1005, 1993) show the same conclusion
under the stronger assumption that an
more >>>

TR05-016 | 13th January 2005
Tomas Feder, Daniel Ford

Classification of Bipartite Boolean Constraint Satisfaction through Delta-Matroid Intersection

Matroid intersection has a known polynomial time algorithm using an
oracle. We generalize this result to delta-matroids that do not have
equality as a restriction, and give a polynomial time algorithm for
delta-matroid intersection on delta-matroids without equality using an
oracle. We note that when equality is present, delta-matroid intersection
more >>>

TR05-017 | 28th January 2005
Phuong Nguyen

Two-Sorted Theories for L, SL, NL and P

We introduce ``minimal'' two--sorted first--order theories VL, VSL, VNL and VP
that characterize the classes L, SL, NL and P in the same
way that Buss's $S^i_2$ hierarchy characterizes the polynomial time hierarchy.
Our theories arise from natural combinatorial problems, namely the st-Connectivity
Problem and the Circuit Value Problem.
It ... more >>>

TR05-018 | 6th February 2005
Oded Goldreich

On Promise Problems (a survey in memory of Shimon Even [1935-2004])

The notion of promise problems was introduced and initially studied
by Even, Selman and Yacobi
(Information and Control, Vol.~61, pages 159-173, 1984).
In this article we survey some of the applications that this
notion has found in the twenty years that elapsed.
These include the notion ... more >>>

TR05-019 | 9th February 2005
Venkatesan Guruswami, Atri Rudra

Tolerant Locally Testable Codes

An error-correcting code is said to be {\em locally testable} if it has an
efficient spot-checking procedure that can distinguish codewords
from strings that are far from every codeword, looking at very few
locations of the input in doing so. Locally testable codes (LTCs) have
generated ... more >>>

TR05-020 | 22nd November 2004
Sourav Chakraborty

On the Sensitivity of Cyclically-Invariant Boolean Functions

In this paper we construct a cyclically invariant Boolean function
whose sensitivity is $\Theta(n^{1/3})$. This result answers two
previously published questions. Tur\'an (1984) asked if any
Boolean function, invariant under some transitive group of
permutations, has sensitivity $\Omega(\sqrt{n})$. Kenyon and Kutin
(2004) asked whether for a ``nice'' function the product ... more >>>

TR05-021 | 14th February 2005
Stasys Jukna

Disproving the single level conjecture

Revisions: 2 , Comments: 1

We consider the minimal number of AND and OR gates in monotone
circuits for quadratic boolean functions, i.e. disjunctions of
length-$2$ monomials. The single level conjecture claims that
monotone single level circuits, i.e. circuits which have only one
level of AND gates, for quadratic functions ... more >>>

TR05-022 | 19th February 2005
Omer Reingold, Luca Trevisan, Salil Vadhan

Pseudorandom Walks in Biregular Graphs and the RL vs. L Problem

Motivated by Reingold's recent deterministic log-space algorithm for Undirected S-T Connectivity (ECCC TR 04-94), we revisit the general RL vs. L question, obtaining the following results.

1. We exhibit a new complete problem for RL: S-T Connectivity restricted to directed graphs for which the random walk is promised to have ... more >>>

TR05-023 | 16th February 2005
Robert H. Sloan, Balázs Szörényi, György Turán

On k-term DNF with largest number of prime implicants

It is known that a k-term DNF can have at most 2^k ? 1 prime implicants and this bound is sharp. We determine all k-term DNF having the maximal number of prime implicants. It is shown that a DNF is maximal if and only if it corresponds to a non-repeating ... more >>>

TR05-024 | 8th February 2005
Michael Bauland, Elmar Böhler, Nadia Creignou, Steffen Reith, Henning Schnoor, Heribert Vollmer

Quantified Constraints: The Complexity of Decision and Counting for Bounded Alternation

We consider constraint satisfaction problems parameterized by the set of allowed constraint predicates. We examine the complexity of quantified constraint satisfaction problems with a bounded number of quantifier alternations and the complexity of the associated counting problems. We obtain classification results that completely solve the Boolean case, and we show ... more >>>

TR05-025 | 20th February 2005
Zeev Dvir, Ran Raz

Analyzing Linear Mergers

Mergers are functions that transform k (possibly dependent)
random sources into a single random source, in a way that ensures
that if one of the input sources has min-entropy rate $\delta$
then the output has min-entropy rate close to $\delta$. Mergers
have proven to be a very useful tool in ... more >>>

TR05-026 | 15th February 2005
Scott Aaronson

NP-complete Problems and Physical Reality

Can NP-complete problems be solved efficiently in the physical universe?
I survey proposals including soap bubbles, protein folding, quantum
computing, quantum advice, quantum adiabatic algorithms,
quantum-mechanical nonlinearities, hidden variables, relativistic time
dilation, analog computing, Malament-Hogarth spacetimes, quantum
gravity, closed timelike curves, and "anthropic computing." The ... more >>>

TR05-027 | 19th February 2005
Daniel Rolf

Derandomization of PPSZ for Unique-$k$-SAT

The PPSZ algorithm presented by Paturi, Pudlak, Saks, and Zane in 1998 has the nice feature that the only satisfying solution of a uniquely satisfiable $3$-SAT formulas can be found in expected running time at most $\Oc(1.3071^n).$ Using the technique of limited independence, we can derandomize this algorithm yielding $\Oc(1.3071^n)$ ... more >>>

TR05-028 | 12th February 2005
Elmar Böhler

On the Lattice of Clones Below the Polynomial Time Functions

A clone is a set of functions that is closed under generalized substitution.
The set FP of functions being computable deterministically in polynomial
time is such a clone. It is well-known that the set of subclones of every
clone forms a lattice. We study the lattice below FP, which ... more >>>

TR05-029 | 2nd March 2005
Frank Neumann, Marco Laumanns

Speeding Up Approximation Algorithms for NP-hard Spanning Forest Problems by Multi-objective Optimization

Revisions: 1

We give faster approximation algorithms for the
generalization of two NP-hard spanning tree problems. First,
we investigate the problem of minimizing the degree of
minimum spanning forests. The task is to compute for each
number of connected components a minimum spanning forest
whose degree is as small as possible. Fischer
more >>>

TR05-030 | 12th February 2005
Evgeny Dantsin, Alexander Wolpert

An Improved Upper Bound for SAT

We give a randomized algorithm for testing satisfiability of Boolean formulas in conjunctive normal form with no restriction on clause length. Its running time is at most $2^{n(1-1/\alpha)}$ up to a polynomial factor, where $\alpha = \ln(m/n) + O(\ln \ln m)$ and $n$, $m$ are respectively the number of variables ... more >>>

TR05-031 | 1st March 2005
Carme Alvarez, Joaquim Gabarro, Maria Serna

Pure Nash equilibria in games with a large number of actions

We study the computational complexity of deciding the existence of a
Pure Nash Equilibrium in multi-player strategic games.
We address two fundamental questions: how can we represent a game?, and
how can we represent a game with polynomial pay-off functions?
Our results show that the computational complexity of
deciding ... more >>>

TR05-032 | 16th March 2005
Gudmund Skovbjerg Frandsen, Peter Bro Miltersen

Reviewing Bounds on the Circuit Size of the Hardest Functions

In this paper we review the known bounds for $L(n)$, the circuit size
complexity of the hardest Boolean function on $n$ input bits. The
best known bounds appear to be $$\frac{2^n}{n}(1+\frac{\log
n}{n}-O(\frac{1}{n})) \leq L(n) \leq\frac{2^n}{n}(1+3\frac{\log
n}{n}+O(\frac{1}{n}))$$ However, the bounds do not seem to be
explicitly stated in the literature. We ... more >>>

TR05-033 | 5th March 2005
Martin Furer, Shiva Prasad Kasiviswanathan

Algorithms for Counting 2-SAT Solutions and Colorings with Applications

An algorithm is presented for counting the number of maximum weight satisfying assignments of a 2SAT formula. The worst case running time of $O(\mbox{poly($n$)} \cdot 1.2461^n)$ for formulas with $n$ variables improves on the previous bound of $O(\mbox{poly($n$)} \cdot 1.2561^n)$ by Dahll\"of, Jonsson, and Wahlstr\"om . The weighted 2SAT counting ... more >>>

TR05-034 | 5th April 2005
Luca Trevisan

Approximation Algorithms for Unique Games

Revisions: 1 , Comments: 1

Khot formulated in 2002 the "Unique Games Conjectures" stating that, for any epsilon > 0, given a system of constraints of a certain form, and the promise that there is an assignment that satisfies a 1-epsilon fraction of constraints, it is intractable to find a solution that satisfies even an ... more >>>

TR05-035 | 24th March 2005
Christian Glaßer, Stephen Travers, Klaus W. Wagner

A Reducibility that Corresponds to Unbalanced Leaf-Language Classes

We introduce the polynomial-time tree reducibility
(ptt-reducibility). Our main result states that for
languages $B$ and $C$ it holds that
$B$ ptt-reduces to $C$ if and only if
the unbalanced leaf-language class of $B$ is robustly contained in
the unbalanced leaf-language class of $C$.
... more >>>

TR05-036 | 28th March 2005
Hubie Chen

Quantified Constraint Satisfaction, Maximal Constraint Languages, and Symmetric Polymorphisms

The constraint satisfaction problem (CSP) is a convenient framework for modelling search problems; the CSP involves deciding, given a set of constraints on variables, whether or not there is an assignment to the variables satisfying all of the constraints. This paper is concerned with the quantified constraint satisfaction problem (QCSP), ... more >>>

TR05-037 | 8th April 2005
Eric Allender, Peter Bürgisser, Johan Kjeldgaard-Pedersen, Peter Bro Miltersen

On the Complexity of Numerical Analysis

Revisions: 1 , Comments: 1

We study two quite different approaches to understanding the complexity
of fundamental problems in numerical analysis. We show that both hinge
on the question of understanding the complexity of the following problem,
which we call PosSLP:
Given a division-free straight-line program
producing an integer N, decide whether N>0.
more >>>

TR05-038 | 10th April 2005
Ran Raz

Quantum Information and the PCP Theorem

We show how to encode $2^n$ (classical) bits $a_1,...,a_{2^n}$
by a single quantum state $|\Psi \rangle$ of size $O(n)$ qubits,
such that:
for any constant $k$ and any $i_1,...,i_k \in \{1,...,2^n\}$,
the values of the bits $a_{i_1},...,a_{i_k}$ can be retrieved
from $|\Psi \rangle$ by a one-round Arthur-Merlin interactive ... more >>>

TR05-039 | 13th April 2005
Irit Dinur, Elchanan Mossel, Oded Regev

Conditional Hardness for Approximate Coloring

We study the approximate-coloring(q,Q) problem: Given a graph G, decide
whether \chi(G) \le q or \chi(G)\ge Q. We derive conditional
hardness for this problem for any constant 3\le q < Q. For q \ge
4, our result is based on Khot's 2-to-1 conjecture [Khot'02].
For q=3, we base our hardness ... more >>>

TR05-040 | 13th April 2005
Scott Aaronson

Oracles Are Subtle But Not Malicious

Theoretical computer scientists have been debating the role of
oracles since the 1970's. This paper illustrates both that oracles
can give us nontrivial insights about the barrier problems in
circuit complexity, and that they need not prevent us from trying to
solve those problems.

First, we ... more >>>

TR05-041 | 12th April 2005
Shengyu Zhang

(Almost) tight bounds for randomized and quantum Local Search on hypercubes and grids

Revisions: 2

The Local Search problem, which finds a
local minimum of a black-box function on a given graph, is of both
practical and theoretical importance to many areas in computer
science and natural sciences. In this paper, we show that for the
Boolean hypercube $\B^n$, the randomized query complexity of Local
more >>>

TR05-042 | 15th April 2005
Lance Fortnow, Adam Klivans

Linear Advice for Randomized Logarithmic Space

Revisions: 1

We show that RL is contained in L/O(n), i.e., any language computable
in randomized logarithmic space can be computed in deterministic
logarithmic space with a linear amount of non-uniform advice. To
prove our result we show how to take an ultra-low space walk on
the Gabber-Galil expander graph.

more >>>

TR05-043 | 5th April 2005
Emanuele Viola

Pseudorandom Bits for Constant-Depth Circuits with Few Arbitrary Symmetric Gates

We exhibit an explicitly computable `pseudorandom' generator stretching $l$ bits into $m(l) = l^{\Omega(\log l)}$ bits that look random to constant-depth circuits of size $m(l)$ with $\log m(l)$ arbitrary symmetric gates (e.g. PARITY, MAJORITY). This improves on a generator by Luby, Velickovic and Wigderson (ISTCS '93) that achieves the same ... more >>>

TR05-044 | 6th April 2005
Zeev Dvir, Amir Shpilka

Locally Decodable Codes with 2 queries and Polynomial Identity Testing for depth 3 circuits

In this work we study two seemingly unrelated notions. Locally Decodable Codes(LDCs) are codes that allow the recovery of each message bit from a constant number of entries of the codeword. Polynomial Identity Testing (PIT) is one of the fundamental problems of algebraic complexity: we are given a circuit computing ... more >>>

TR05-045 | 12th April 2005
Philippe Moser

Martingale Families and Dimension in P

Revisions: 1

We introduce a new measure notion on small complexity classes (called F-measure), based on martingale families,
that get rid of some drawbacks of previous measure notions:
martingale families can make money on all strings,
and yield random sequences with an equal frequency of 0's and 1's.
As applications to F-measure,
more >>>

TR05-046 | 17th April 2005
Irit Dinur

The PCP theorem by gap amplification

Revisions: 1 , Comments: 3

Let C={c_1,...,c_n} be a set of constraints over a set of
variables. The {\em satisfiability-gap} of C is the smallest
fraction of unsatisfied constraints, ranging over all possible
assignments for the variables.

We prove a new combinatorial amplification lemma that doubles the
satisfiability-gap of a constraint-system, with only a linear ... more >>>

TR05-047 | 10th April 2005
Kooshiar Azimian, Mahmoud Salmasizadeh, Javad Mohajeri

Weak Composite Diffie-Hellman is not Weaker than Factoring

In1985, Shmuley proposed a theorem about intractability of Composite Diffie-Hellman [Sh85]. The Theorem of Shmuley may be paraphrased as saying that if there exist a probabilistic poly-time oracle machine which solves the Diffie-Hellman modulo an
RSA-number with odd-order base then there exist a probabilistic algorithm which factors the modulo. ... more >>>

TR05-048 | 11th April 2005
Moti Yung, Yunlei Zhao

Constant-Round Concurrently-Secure rZK in the (Real) Bare Public-Key Model

Revisions: 3

We present constant-round concurrently secure (sound) resettable
zero-knowledge (rZK-CS) arguments in the bare public-key (BPK)
model. Our constructions deal with general NP ZK-arguments as well
as with highly efficient ZK-arguments for number-theoretic
languages, most relevant to identification scenarios. These are the
first constant-round protocols of ... more >>>

TR05-049 | 1st April 2005
Joan Boyar, rene peralta

The Exact Multiplicative Complexity of the Hamming Weight Function

We consider the problem of computing the Hamming weight of an n-bit vector using a circuit with gates for GF2 addition and multiplication only. We show the number of multiplications necessary and sufficient to build such a circuit is n - |n| where |n| is the Hamming weight of the ... more >>>

TR05-050 | 18th April 2005
Uriel Feige, Eran Ofek

Finding a Maximum Independent Set in a Sparse Random Graph

Revisions: 1

We consider the problem of finding a maximum independent set in a
random graph. The random graph $G$ is modelled as follows. Every
edge is included independently with probability $\frac{d}{n}$, where
$d$ is some sufficiently large constant. Thereafter, for some
constant $\alpha$, a subset $I$ of $\alpha n$ vertices is ... more >>>

TR05-051 | 18th March 2005
Predrag Tosic

On Complexity of Counting Fixed Points in Certain Classes of Graph Automata

Revisions: 2

We study the computational complexity of counting the fixed point configurations in certain discrete dynamical systems. We prove that both exact and approximate counting in Sequential and Synchronous Dynamical Systems (SDSs and SyDS, respectrively) is computationally intractable, even when each node is required to update according to a symmetric Boolean ... more >>>

TR05-052 | 5th May 2005
Grant Schoenebeck, Salil Vadhan

The Computational Complexity of Nash Equilibria in Concisely Represented Games

Games may be represented in many different ways, and different representations of games affect the complexity of problems associated with games, such as finding a Nash equilibrium. The traditional method of representing a game is to explicitly list all the payoffs, but this incurs an exponential blowup as the number ... more >>>

TR05-053 | 4th May 2005
Paul Beame, Nathan Segerlind

Lower bounds for Lovasz-Schrijver systems and beyond follow from multiparty communication complexity

We prove that an \omega(log^3 n) lower bound for the three-party number-on-the-forehead (NOF) communication complexity of the set-disjointness function implies an n^\omega(1) size lower bound for tree-like Lovasz-Schrijver systems that refute unsatisfiable CNFs. More generally, we prove that an n^\Omega(1) lower bound for the (k+1)-party NOF communication complexity of set-disjointness ... more >>>

TR05-054 | 19th May 2005
Konstantin Pervyshev

Time Hierarchies for Computations with a Bit of Advice

A polynomial time hierarchy for ZPTime with one bit of advice is proved. That is for any constants a and b such that 1 < a < b, ZPTime[n^a]/1 \subsetneq ZPTime[n^b]/1.

The technique introduced in this paper is very general and gives the same hierarchy for NTime \cap coNTime, UTime, ... more >>>

TR05-055 | 19th May 2005
Bruno Codenotti, Amin Saberi, Kasturi Varadarajan, Yinyu Ye

Leontief Economies Encode Nonzero Sum Two-Player Games

We give a reduction from any two-player game to a special case of
the Leontief exchange economy, with the property that the Nash equilibria of the game and the
equilibria of the market are in one-to-one correspondence.

Our reduction exposes a potential hurdle inherent in solving certain
families of market ... more >>>

TR05-056 | 25th April 2005
Alexis Kaporis, Efpraxia Politopoulou, Paul Spirakis

The Price of Optimum in Stackelberg Games

Consider a system M of parallel machines, each with a strictly increasing and differentiable load dependent latency function. The users of such a system are of infinite number and act selfishly, routing their infinitesimally small portion of the total flow r they control, to machines of currently minimum delay. It ... more >>>

TR05-057 | 19th May 2005
Venkatesan Guruswami, Valentine Kabanets

Hardness amplification via space-efficient direct products

We prove a version of the derandomized Direct Product Lemma for
deterministic space-bounded algorithms. Suppose a Boolean function
$g:\{0,1\}^n\to\{0,1\}$ cannot be computed on more than $1-\delta$
fraction of inputs by any deterministic time $T$ and space $S$
algorithm, where $\delta\leq 1/t$ for some $t$. Then, for $t$-step
walks $w=(v_1,\dots, v_t)$ ... more >>>

TR05-058 | 24th May 2005
Sanjeev Arora, Eli Berger, Elad Hazan, Guy Kindler, Muli Safra

On Non-Approximability for Quadratic Programs

This paper studies the computational complexity of the following type of
quadratic programs: given an arbitrary matrix whose diagonal elements are zero, find $x \in \{-1,+1\}^n$ that maximizes $x^TA x$. This problem recently attracted attention due to its application in various clustering settings (Charikar and Wirth, 2004) as well as ... more >>>

TR05-059 | 9th May 2005
Víctor Dalmau, Ricard Gavaldà, Pascal Tesson, Denis Thérien

Tractable Clones of Polynomials over Semigroups

It is well known that coset-generating relations lead to tractable
constraint satisfaction problems. These are precisely the relations closed
under the operation $xy^{-1}z$ where the multiplication is taken in
some finite group. Bulatov et al. have on the other hand shown that
any clone containing the multiplication of some ``block-group'' ... more >>>

TR05-060 | 30th May 2005
Philippe Moser

Generic Density and Small Span Theorem

We refine the genericity concept of Ambos-Spies et al, by assigning a real number in $[0,1]$ to every generic set, called its generic density.
We construct sets of generic density any E-computable real in $[0,1]$.
We also introduce strong generic density, and show that it is related to packing ... more >>>

TR05-061 | 15th June 2005
Ronen Gradwohl, Guy Kindler, Omer Reingold, Amnon Ta-Shma

On the Error Parameter of Dispersers

Optimal dispersers have better dependence on the error than
optimal extractors. In this paper we give explicit disperser
constructions that beat the best possible extractors in some
parameters. Our constructions are not strong, but we show that
having such explicit strong constructions implies a solution
to the Ramsey graph construction ... more >>>

TR05-062 | 17th June 2005
A. Pavan, Vinodchandran Variyam

2-Local Random Reductions to 3-Valued Functions

Yao (in a lecture at DIMACS Workshop on structural complexity and
cryptography) showed that if a language L is 2-locally-random
reducible to a Boolean functio, then L is in PSPACE/poly.
Fortnow and Szegedy quantitatively improved Yao's result to show that
such languages are in fact in NP/poly (Information Processing Letters, ... more >>>

TR05-063 | 24th June 2005
Bodo Manthey, Rüdiger Reischuk

Smoothed Analysis of the Height of Binary Search Trees

Revisions: 2

Binary search trees are one of the most fundamental data structures. While the
height of such a tree may be linear in the worst case, the average height with
respect to the uniform distribution is only logarithmic. The exact value is one
of the best studied problems in average case ... more >>>

TR05-064 | 26th June 2005
Howard Karloff, Subhash Khot, Aranyak Mehta, Yuval Rabani

On earthmover distance, metric labeling, and 0-extension

We study the classification problem {\sc Metric Labeling} and its special case {\sc 0-Extension} in the context of earthmover metrics. Researchers recently proposed using earthmover metrics to get a polynomial time-solvable relaxation of {\sc Metric Labeling}; until now, however, no one knew if the integrality ratio was constant or not, ... more >>>

TR05-065 | 26th June 2005
Alexander Barvinok, Alex Samorodnitsky

Random Weighting, Asymptotic Counting, and Inverse Isoperimetry

For a family X of k-subsets of the set 1...n, let |X| be the cardinality of X and let Gamma(X, mu) be the expected maximum weight of a subset from X when the weights of 1...n are chosen independently at random from a symmetric probability distribution mu on R. We ... more >>>

TR05-066 | 4th June 2005
Jakob Nordström

Narrow Proofs May Be Spacious: Separating Space and Width in Resolution

Revisions: 2 , Comments: 1

The width of a resolution proof is the maximal number of literals in any clause of the proof. The space of a proof is the maximal number of memory cells used if the proof is only allowed to resolve on clauses kept in memory. Both of these measures have previously ... more >>>

TR05-067 | 28th June 2005
Zeev Dvir, Amir Shpilka

An Improved Analysis of Mergers

Mergers are functions that transform k (possibly dependent) random sources into a single random source, in a way that ensures that if one of the input sources has min-entropy rate $\delta$ then the output has min-entropy rate close to $\delta$. Mergers have proven to be a very useful tool in ... more >>>

TR05-068 | 7th July 2005
Christian Glaßer, A. Pavan, Alan L. Selman, Liyu Zhang

Redundancy in Complete Sets

We show that a set is m-autoreducible if and only if it is m-mitotic. This solves a long standing open question in a surprising way. As a consequence of this unconditional result and recent work by Glasser et al., complete sets for all of the following complexity classes are m-mitotic: ... more >>>

TR05-069 | 11th July 2005
Piotr Berman, Marek Karpinski

8/7-Approximation Algorithm for (1,2)-TSP

Revisions: 2

We design a polynomial time 8/7-approximation algorithm for the Traveling Salesman Problem in which all distances are either one or two. This improves over the best known approximation factor of 7/6 for that problem. As a direct application we get a 7/6-approximation algorithm for the Maximum Path Cover Problem, similarily ... more >>>

TR05-070 | 6th July 2005
Mahdi Cheraghchi

On Matrix Rigidity and the Complexity of Linear Forms

The rigidity function of a matrix is defined as the minimum number of its entries that need to be changed in order to reduce the rank of the matrix to below a given parameter. Proving a strong enough lower bound on the rigidity of a matrix implies a nontrivial lower ... more >>>

TR05-071 | 29th June 2005
Marius Zimand

Simple extractors via constructions of cryptographic pseudo-random generators

Trevisan has shown that constructions of pseudo-random generators from
hard functions (the Nisan-Wigderson approach) also produce extractors.
We show that constructions of pseudo-random generators from one-way permutations
(the Blum-Micali-Yao approach) can be used for building extractors as well.
Using this new technique we build extractors that ... more >>>

TR05-072 | 11th July 2005
Christian Glaßer, Alan L. Selman, Liyu Zhang

Survey of Disjoint NP-Pairs and Relations to Propositional Proof Systems

We survey recent results on disjoint NP-pairs. In particular, we survey the relationship of disjoint NP-pairs to the theory of proof systems for propositional calculus.

more >>>

TR05-073 | 14th July 2005
Oded Goldreich, Dana Ron

Approximating Average Parameters of Graphs.

Inspired by Feige ({\em 36th STOC}, 2004),
we initiate a study of sublinear randomized algorithms
for approximating average parameters of a graph.
Specifically, we consider the average degree of a graph
and the average distance between pairs of vertices in a graph.
Since our focus is on sublinear algorithms, ... more >>>

TR05-074 | 8th June 2005
Li-Sha Huang, Xiaotie Deng

On Complexity of Market Equilibria with Maximum Social Welfare

We consider the computational complexity of the market equilibrium
problem by exploring the structural properties of the Leontief
exchange economy. We prove that, for economies guaranteed to have
a market equilibrium, finding one with maximum social welfare or
maximum individual welfare is NP-hard. In addition, we prove that
counting the ... more >>>

TR05-075 | 4th July 2005
Martin Dyer, Leslie Ann Goldberg, Mark Jerrum

Dobrushin conditions and Systematic Scan

Revisions: 1

We consider Glauber dynamics on finite spin systems.
The mixing time of Glauber dynamics can be bounded
in terms of the influences of sites on each other.
We consider three parameters bounding these influences ---
$\alpha$, the total influence on a site, as studied by Dobrushin;
$\alpha'$, the total influence ... more >>>

TR05-076 | 2nd July 2005
Dima Grigoriev, Edward Hirsch, Konstantin Pervyshev

Time hierarchies for cryptographic function inversion with advice

We prove a time hierarchy theorem for inverting functions
computable in polynomial time with one bit of advice.
In particular, we prove that if there is a strongly
one-way function, then for any k and for any polynomial p,
there is a function f computable in linear time
with one ... more >>>

TR05-077 | 15th July 2005
Zenon Sadowski

On a D-N-optimal acceptor for TAUT

The notion of an optimal acceptor for TAUT (the optimality
property is stated only for input strings from TAUT) comes from the line
of research aimed at resolving the question of whether optimal
propositional proof systems exist. In this paper we introduce two new
types of optimal acceptors, a D-N-optimal ... more >>>

TR05-078 | 25th May 2005
Kooshiar Azimian, Javad Mohajeri, Mahmoud Salmasizadeh, Siamak Fayyaz

A Verifiable Partial Key Escrow, Based on McCurley Encryption Scheme

Revisions: 1

In this paper, firstly we propose two new concepts concerning the notion of key escrow encryption schemes: provable partiality and independency. Roughly speaking we say that a scheme has provable partiality if existing polynomial time algorithm for recovering the secret knowing escrowed information implies a polynomial time algorithm that can ... more >>>

TR05-079 | 25th July 2005
Stasys Jukna

Expanders and time-restricted branching programs

The \emph{replication number} of a branching program is the minimum
number R such that along every accepting computation at most R
variables are tested more than once. Hence 0\leq R\leq n for every
branching program in n variables. The best results so far were
exponential ... more >>>

TR05-080 | 21st July 2005
Michael R. Fellows, Frances A. Rosamond, Udi Rotics, Stefan Szeider

Proving NP-hardness for clique-width I: non-approximability of sequential clique-width

Revisions: 1

Clique-width is a graph parameter, defined by a composition mechanism
for vertex-labeled graphs, which measures in a certain sense the
complexity of a graph. Hard graph problems (e.g., problems
expressible in Monadic Second Order Logic, that includes NP-hard
problems) can be solved efficiently for graphs of certified small
clique-width. It ... more >>>

TR05-081 | 21st July 2005
Michael R. Fellows, Frances A. Rosamond, Udi Rotics, Stefan Szeider

Proving NP-hardness for clique-width II: non-approximability of clique-width

Revisions: 1

Clique-width is a graph parameter that measures in a certain sense the
complexity of a graph. Hard graph problems (e.g., problems
expressible in Monadic Second Order Logic with second-order
quantification on vertex sets, that includes NP-hard problems) can be
solved efficiently for graphs of certified small clique-width. It is
widely ... more >>>

TR05-082 | 3rd June 2005
Jorge Castro

On the Query Complexity of Quantum Learners

This paper introduces a framework for quantum exact learning via queries, the so-called quantum protocol. It is shown that usual protocols in the classical learning setting have quantum counterparts. A combinatorial notion, the general halving dimension, is also introduced. Given a quantum protocol and a target concept class, the general ... more >>>

TR05-083 | 24th July 2005
Olaf Beyersdorff

Disjoint NP-Pairs from Propositional Proof Systems

For a proof system P we introduce the complexity class DNPP(P)
of all disjoint NP-pairs for which the disjointness of the pair is
efficiently provable in the proof system P.
We exhibit structural properties of proof systems which make the
previously defined canonical NP-pairs of these proof systems hard ... more >>>

TR05-084 | 31st July 2005
Mickey Brautbar, Alex Samorodnitsky

Approximating the entropy of large alphabets

We consider the problem of approximating the entropy of a discrete distribution P on a domain of size q, given access to n independent samples from the distribution. It is known that n > q is necessary, in general, for a good additive estimate of the entropy. A problem of ... more >>>

TR05-085 | 5th August 2005
Asaf Shapira, Noga Alon

Homomorphisms in Graph Property Testing - A Survey

Property-testers are fast randomized algorithms for distinguishing
between graphs (and other combinatorial structures) satisfying a
certain property, from those that are far from satisfying it. In
many cases one can design property-testers whose running time is in
fact {\em independent} of the size of the input. In this paper we
more >>>

TR05-086 | 14th August 2005
Dana Moshkovitz, Ran Raz

Sub-Constant Error Low Degree Test of Almost Linear Size

Revisions: 1

Given a function f:F^m \rightarrow F over a finite
field F, a low degree tester tests its proximity to
an m-variate polynomial of total degree at most d
over F. The tester is usually given access to an oracle
A providing the supposed restrictions of f to
affine subspaces of ... more >>>

TR05-087 | 9th August 2005
Alexander Healy, Emanuele Viola

Constant-Depth Circuits for Arithmetic in Finite Fields of Characteristic Two

We study the complexity of arithmetic in finite fields of characteristic two, $\F_{2^n}$.
We concentrate on the following two problems:

Iterated Multiplication: Given $\alpha_1, \alpha_2,..., \alpha_t \in \F_{2^n}$, compute $\alpha_1 \cdot \alpha_2 \cdots \alpha_t \in \F_{2^n}$.

Exponentiation: Given $\alpha \in \F_{2^n}$ and a $t$-bit integer $k$, compute $\alpha^k \in \F_{2^n}$.

... more >>>

TR05-088 | 3rd August 2005
Jan Arpe

Learning Juntas in the Presence of Noise

The combination of two major challenges in machine learning is investigated: dealing with large amounts of irrelevant information and learning from noisy data. It is shown that large classes of Boolean concepts that depend on a small number of variables---so-called juntas---can be learned efficiently from random examples corrupted by random ... more >>>

TR05-089 | 30th July 2005
Xiaoyang Gu, Jack H. Lutz, Philippe Moser

Dimensions of Copeland-Erdos Sequences

The base-$k$ {\em Copeland-Erd\"os sequence} given by an infinite
set $A$ of positive integers is the infinite
sequence $\CE_k(A)$ formed by concatenating the base-$k$
representations of the elements of $A$ in numerical
order. This paper concerns the following four
The {\em finite-state dimension} $\dimfs (\CE_k(A))$,
a finite-state ... more >>>

TR05-090 | 17th August 2005
Paul Goldberg, Christos H. Papadimitriou

Reducibility Among Equilibrium Problems

We address the fundamental question of whether the Nash equilibria of
a game can be computed in polynomial time. We describe certain
efficient reductions between this problem for
normal form games with a fixed number of players
and graphical games with fixed degree. Our main result is that ... more >>>

TR05-091 | 11th August 2005
Predrag Tosic

Counting Fixed Points and Gardens of Eden of Sequential Dynamical Systems on Planar Bipartite Graphs

Revisions: 1

We study counting various types of con gurations in certain classes of graph
automata viewed as discrete dynamical systems. The graph automata models
of our interest are Sequential and Synchronous Dynamical Systems (SDSs and
SyDSs, respectively). These models have been proposed as a mathematical foun-
dation for a theory of ... more >>>

TR05-092 | 23rd August 2005
Eyal Rozenman, Salil Vadhan

Derandomized Squaring of Graphs

We introduce a "derandomized" analogue of graph squaring. This
operation increases the connectivity of the graph (as measured by the
second eigenvalue) almost as well as squaring the graph does, yet only
increases the degree of the graph by a constant factor, instead of
squaring the degree.

One application of ... more >>>

TR05-093 | 24th August 2005
Daniele Micciancio, Shien Jin Ong, Amit Sahai, Salil Vadhan

Concurrent Zero Knowledge without Complexity Assumptions

We provide <i>unconditional</i> constructions of <i>concurrent</i>
statistical zero-knowledge proofs for a variety of non-trivial
problems (not known to have probabilistic polynomial-time
algorithms). The problems include Graph Isomorphism, Graph
Nonisomorphism, Quadratic Residuosity, Quadratic Nonresiduosity, a
restricted version of Statistical Difference, and approximate
versions of the (<b>coNP</b> forms of the) Shortest Vector ... more >>>

TR05-094 | 9th August 2005
Michal Parnas, Dana Ron

On Approximating the Minimum Vertex Cover in Sublinear Time and the Connection to Distributed Algorithms

Revisions: 1

We consider the problem of estimating the size, $VC(G)$, of a
minimum vertex cover of a graph $G$, in sublinear time,
by querying the incidence relation of the graph. We say that
an algorithm is an $(\alpha,\eps)$-approximation algorithm
if it outputs with high probability an estimate $\widehat{VC}$
such that ... more >>>

TR05-095 | 26th August 2005
Noga Alon, Ilan Newman, Alexander Shen, Gábor Tardos, Nikolay Vereshchagin

Partitioning multi-dimensional sets in a small number of ``uniform'' parts

Our main result implies the following easily formulated statement. The set of edges E of every finite bipartite graph can be split into poly(log |E|) subsets so the all the resulting bipartite graphcs are almost regular. The latter means that the ratio between the maximal and minimal non-zero degree of ... more >>>

TR05-096 | 26th August 2005
Boaz Barak, Amit Sahai

How To Play Almost Any Mental Game Over The Net --- Concurrent Composition via Super-Polynomial Simulation

We construct a secure protocol for any multi-party functionality
that remains secure (under a relaxed definition of security) when
executed concurrently with multiple copies of itself and other
protocols. We stress that we do *not* use any assumptions on
existence of trusted parties, common reference string, honest
majority or synchronicity ... more >>>

TR05-097 | 31st August 2005
Jens Groth, Rafail Ostrovsky, Amit Sahai

Perfect Non-Interactive Zero Knowledge for NP

Non-interactive zero-knowledge (NIZK) systems are
fundamental cryptographic primitives used in many constructions,
including CCA2-secure cryptosystems, digital signatures, and various
cryptographic protocols. What makes them especially attractive, is
that they work equally well in a concurrent setting, which is
notoriously hard for interactive zero-knowledge protocols. However,
while for interactive zero-knowledge we ... more >>>

TR05-098 | 4th September 2005
Oded Goldreich

Bravely, Moderately: A Common Theme in Four Recent Results

We highlight a common theme in four relatively recent works
that establish remarkable results by an iterative approach.
Starting from a trivial construct,
each of these works applies an ingeniously designed
sequence of iterations that yields the desired result,
which is highly non-trivial. Furthermore, in each iteration,
more >>>

TR05-099 | 9th September 2005
Leslie G. Valiant

Holographic Algorithms

Complexity theory is built fundamentally on the notion of efficient
reduction among computational problems. Classical
reductions involve gadgets that map solution fragments of one problem to
solution fragments of another in one-to-one, or
possibly one-to-many, fashion. In this paper we propose a new kind of
reduction that allows for gadgets ... more >>>

TR05-100 | 30th August 2005
David Zuckerman

Linear Degree Extractors and the Inapproximability of Max Clique and Chromatic Number

A randomness extractor is an algorithm which extracts randomness from a low-quality random source, using some additional truly random bits. We construct new extractors which require only log n + O(1) additional random bits for sources with constant entropy rate. We further construct dispersers, which are similar to one-sided extractors, ... more >>>

TR05-101 | 20th September 2005
Guy Kindler, Ryan O'Donnell, Subhash Khot, Elchanan Mossel

Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs?

In this paper we show a reduction from the Unique Games problem to the problem of approximating MAX-CUT to within a factor of $\GW + \eps$, for all $\eps > 0$; here $\GW \approx .878567$ denotes the approximation ratio achieved by the Goemans-Williamson algorithm~\cite{GW95}. This implies that if the Unique ... more >>>

TR05-102 | 13th September 2005
Evgeny Dantsin, Edward Hirsch, Alexander Wolpert

Clause Shortening Combined with Pruning Yields a New Upper Bound for Deterministic SAT Algorithms

We give a deterministic algorithm for testing satisfiability of formulas in conjunctive normal form with no restriction on clause length. Its upper bound on the worst-case running time matches the best known upper bound for randomized satisfiability-testing algorithms [Dantsin and Wolpert, SAT 2005]. In comparison with the randomized algorithm in ... more >>>

TR05-103 | 17th August 2005
Leonid Gurvits

A proof of hyperbolic van der Waerden conjecture : the right generalization is the ultimate simplification

Consider a homogeneous polynomial $p(z_1,...,z_n)$ of degree $n$ in $n$ complex variables .
Assume that this polynomial satisfies the property : \\

$|p(z_1,...,z_n)| \geq \prod_{1 \leq i \leq n} Re(z_i)$ on the domain $\{(z_1,...,z_n) : Re(z_i) \geq 0 , 1 \leq i \leq n \}$ . \\

We prove that ... more >>>

TR05-104 | 16th September 2005
Don Coppersmith, Atri Rudra

On the Robust Testability of Product of Codes

Ben-Sasson and Sudan in~\cite{BS04} asked if the following test
is robust for the tensor product of a code with another code--
pick a row (or column) at random and check if the received word restricted to the picked row (or column) belongs to the corresponding code. Valiant showed that ... more >>>

TR05-105 | 24th September 2005
Lance Fortnow, John Hitchcock, A. Pavan, Vinodchandran Variyam, Fengming Wang

Extracting Kolmogorov Complexity with Applications to Dimension Zero-One Laws

We apply recent results on extracting randomness from independent
sources to ``extract'' Kolmogorov complexity. For any $\alpha,
\epsilon > 0$, given a string $x$ with $K(x) > \alpha|x|$, we show
how to use a constant number of advice bits to efficiently
compute another string $y$, $|y|=\Omega(|x|)$, with $K(y) >
(1-\epsilon)|y|$. ... more >>>

TR05-106 | 26th September 2005
Anup Rao

Extractors for a Constant Number of Polynomial Min-Entropy Independent Sources

Revisions: 1

We consider the problem of bit extraction from independent sources. We
construct an extractor that can extract from a constant number of
independent sources of length $n$, each of which have min-entropy
$n^\gamma$ for an arbitrarily small constant $\gamma > 0$. Our
constructions are different from recent extractor constructions
more >>>

TR05-107 | 28th September 2005
Avi Wigderson, David Xiao

A Randomness-Efficient Sampler for Matrix-valued Functions and Applications

Revisions: 1

In this paper we give a randomness-efficient sampler for matrix-valued functions. Specifically, we show that a random walk on an expander approximates the recent Chernoff-like bound for matrix-valued functions of Ahlswede and Winter, in a manner which depends optimally on the spectral gap. The proof uses perturbation theory, and is ... more >>>

TR05-108 | 28th September 2005
Ariel Gabizon, Ran Raz

Deterministic Extractors for Affine Sources over Large Fields

An $(n,k)$-affine source over a finite field $F$ is a random
variable $X=(X_1,...,X_n) \in F^n$, which is uniformly
distributed over an (unknown) $k$-dimensional affine subspace of $
F^n$. We show how to (deterministically) extract practically all
the randomness from affine sources, for any field of size larger
than $n^c$ (where ... more >>>

TR05-109 | 28th September 2005
Ariel Gabizon, Ran Raz, Ronen Shaltiel

Deterministic Extractors for Bit-fixing Sources by Obtaining an Independent Seed

An $(n,k)$-bit-fixing source is a distribution $X$ over $\B^n$ such that
there is a subset of $k$ variables in $X_1,\ldots,X_n$ which are uniformly
distributed and independent of each other, and the remaining $n-k$ variables
are fixed. A deterministic bit-fixing source extractor is a function $E:\B^n
\ar \B^m$ which on ... more >>>

TR05-110 | 3rd October 2005
Saurabh Sanghvi, Salil Vadhan

The Round Complexity of Two-Party Random Selection

We study the round complexity of two-party protocols for
generating a random $n$-bit string such that the output is
guaranteed to have bounded bias (according to some measure) even
if one of the two parties deviates from the protocol (even using
unlimited computational resources). Specifically, we require that
the output's ... more >>>

TR05-111 | 3rd October 2005
Dieter van Melkebeek, Konstantin Pervyshev

A Generic Time Hierarchy for Semantic Models With One Bit of Advice

We show that for any reasonable semantic model of computation and for
any positive integer $a$ and rationals $1 \leq c < d$, there exists a language
computable in time $n^d$ with $a$ bits of advice but not in time $n^c$
with $a$ bits of advice. A semantic ... more >>>

TR05-112 | 12th September 2005
Eran Ofek

On the expansion of the giant component in percolated $(n,d,\lambda)$ graphs

Revisions: 1

Let $d \geq d_0$ be a sufficiently large constant. A $(n,d,c
\sqrt{d})$ graph $G$ is a $d$ regular graph over $n$ vertices whose
second largest eigenvalue (in absolute value) is at most $c
\sqrt{d}$. For any $0 < p < 1, ~G_p$ is the graph induced by
retaining each edge ... more >>>

TR05-113 | 12th September 2005
Bernhard Fuchs

On the Hardness of Range Assignment Problems

We investigate the computational hardness of the {\sc Connectivity},
the {\sc Strong Connectivity} and the {\sc Broadcast} type of Range
Assignment Problems in $\R^2$ and $\R^3$.
We present new reductions for the {\sc Connectivity} problem, which
are easily adapted to suit the other two problems. All reductions
are considerably simpler ... more >>>

TR05-114 | 9th October 2005
Boaz Barak, Shien Jin Ong, Salil Vadhan

Derandomization in Cryptography

We give two applications of Nisan--Wigderson-type ("non-cryptographic") pseudorandom generators in cryptography. Specifically, assuming the existence of an appropriate NW-type generator, we construct:

A one-message witness-indistinguishable proof system for every language in NP, based on any trapdoor permutation. This proof system does not assume a shared random string or any ... more >>>

TR05-115 | 27th September 2005
Constantinos Daskalakis, Paul Goldberg, Christos H. Papadimitriou

The complexity of computing a Nash equilibrium

We resolve the question of the complexity of Nash equilibrium by
showing that the problem of computing a Nash equilibrium in a game
with 4 or more players is complete for the complexity class PPAD.
Our proof uses ideas from the recently-established equivalence
between polynomial-time solvability of normal-form games and
more >>>

TR05-116 | 12th October 2005
Alex Samorodnitsky, Luca Trevisan

Gowers Uniformity, Influence of Variables, and PCPs

Gowers introduced, for d\geq 1, the notion of dimension-d uniformity U^d(f)
of a function f: G -> \C, where G is a finite abelian group and \C are the
complex numbers. Roughly speaking, if a function has small Gowers uniformity
of dimension d, then it ``looks random'' on ... more >>>

TR05-117 | 17th September 2005
Piotr Indyk, David P. Woodruff

Polylogarithmic Private Approximations and Efficient Matching

A private approximation of a function f is defined to be another function F that approximates f in the usual sense, but does not reveal any information about the input x other than what can be deduced from f(x). We give the first two-party private approximation of the Euclidean distance ... more >>>

TR05-118 | 16th October 2005
Jin-Yi Cai, Vinay Choudhary

Valiant's Holant Theorem and Matchgate Tensors

We propose matchgate tensors as a natural and proper language
to develop Valiant's new theory of Holographic Algorithms.
We give a treatment of the central theorem in this theory---the Holant
Theorem---in terms of matchgate tensors.
Some generalizations are presented.

more >>>

TR05-119 | 15th September 2005
Nadia Creignou, Phokion G. Kolaitis, Bruno Zanuttini

Preferred representations of Boolean relations

We introduce the notion of a plain basis for a co-clone in Post's lattice. Such a basis is a set of relations B such that every constraint C over a relation in the co-clone is logically equivalent to a conjunction of equalities and constraints over B and the same variables ... more >>>

TR05-120 | 14th October 2005
Sashka Davis, Russell Impagliazzo

Models of Greedy Algorithms for Graph Problems

Borodin, Nielsen, and Rackoff gave a model of greedy-like algorithms for scheduling problems and Angelopoulos and Borodin extended their work to facility location and set cover problems. We generalize their notion to include other optimization problems, and apply the generalized framework to graph problems. Our goal is to define an ... more >>>

TR05-121 | 17th October 2005
Martin Dyer, Leslie Ann Goldberg, Michael S. Paterson

On counting homomorphisms to directed acyclic graphs

We give a dichotomy theorem for the problem of counting homomorphisms to
directed acyclic graphs. $H$ is a fixed directed acyclic graph.
The problem is, given an input digraph $G$, how many homomorphisms are there
from $G$ to $H$. We give a graph-theoretic classification, showing that
for some digraphs $H$, ... more >>>

TR05-122 | 31st October 2005
Pavel Pudlak

A nonlinear bound on the number of wires in bounded depth circuits

We shall prove a lower bound on the number of edges in some bounded
depth graphs. This theorem is stronger than lower bounds proved on
bounded depth superconcentrators and enables us to prove a lower bound
on certain bounded depth circuits for which we cannot use
superconcentrators: we prove that ... more >>>

TR05-123 | 25th October 2005
Olaf Beyersdorff

Tuples of Disjoint NP-Sets

Disjoint NP-pairs are a well studied complexity theoretic concept with
important applications in cryptography and propositional proof

In this paper we introduce a natural generalization of the notion of
disjoint NP-pairs to disjoint k-tuples of NP-sets for k>1.
We define subclasses of ... more >>>

TR05-124 | 2nd November 2005
Kooshiar Azimian

Breaking Diffie-Hellman is no Easier than Root Finding

In this paper we compare hardness of two well known problems: the Diffie-Hellman problem and the root finding problem. We prove that in any cyclic group computing Diffie-Hellman is not weaker than root finding if certain circumstances are met. As will be discussed in the paper this theorem can affect ... more >>>

TR05-125 | 2nd November 2005
Sofya Raskhodnikova, Dana Ron, Ronitt Rubinfeld, Amir Shpilka, Adam Smith

Sublinear Algorithms for Approximating String Compressibility and the Distribution Support Size

We raise the question of approximating compressibility of a string with respect to a fixed compression scheme, in sublinear time. We study this question in detail for two popular lossless compression schemes: run-length encoding (RLE) and Lempel-Ziv (LZ), and present algorithms and lower bounds for approximating compressibility with respect to ... more >>>

TR05-126 | 5th November 2005
Eric Allender, Lisa Hellerstein, Paul McCabe, Michael Saks

Minimizing DNF Formulas and AC0 Circuits Given a Truth Table

For circuit classes R, the fundamental computational problem, Min-R,
asks for the minimum R-size of a boolean function presented as a truth
table. Prominent examples of this problem include Min-DNF, and
Min-Circuit (also called MCSP). We begin by presenting a new reduction
proving that Min-DNF is NP-complete. It is significantly ... more >>>

TR05-127 | 5th November 2005
Vitaly Feldman

Hardness of Approximate Two-level Logic Minimization and PAC Learning with Membership Queries

Revisions: 1

Producing a small DNF expression consistent with given data is a
classical problem in computer science that occurs in a number of forms and
has numerous applications. We consider two standard variants of this
problem. The first one is two-level logic minimization or finding a minimal
more >>>

TR05-128 | 27th October 2005
Miroslava Sotáková

The normal form of reversible circuits consisting of CNOT and NOT gates

This paper deals with the reversible circuits with n input and
output nodes, consisting of the reversible gates FAN-IN=FAN-OUT<3.
We define a normal form of such type of circuits and describe a reduction
algorithm to transform a circuit in this form. Furthermore we use it for
checking whether two circuits ... more >>>

TR05-129 | 30th October 2005
Scott Aaronson

QMA/qpoly Is Contained In PSPACE/poly: De-Merlinizing Quantum Protocols

This paper introduces a new technique for removing existential quantifiers
over quantum states. Using this technique, we show that there is no way
to pack an exponential number of bits into a polynomial-size quantum
state, in such a way that the value of any one of those bits ... more >>>

TR05-130 | 31st October 2005
Ahuva Mu'alem

A Note on Testing Truthfulness

This work initiates the study of algorithms
for the testing of monotonicity of mechanisms.
Such testing algorithms are useful for
searching dominant strategy mechanisms.
An $\e$-tester for monotonicity
is given a query access to a mechanism,
accepts if monotonicity is satisfied,
and rejects with high probability if more than $\e$-fraction
more >>>

TR05-131 | 7th November 2005
Don Coppersmith, Lisa Fleischer, Atri Rudra

Ordering by weighted number of wins gives a good ranking for weighted tournaments

We consider the following simple algorithm for feedback arc set problem in weighted tournaments --- order the vertices by their weighted indegrees. We show that this algorithm has an approximation guarantee of $5$ if the weights satisfy \textit{probability constraints}
(for any pair of vertices $u$ and $v$, $w_{uv}+w_{vu}=1$). Special cases ... more >>>

TR05-132 | 8th November 2005
Venkatesan Guruswami

Algebraic-geometric generalizations of the Parvaresh-Vardy codes

This paper is concerned with a new family of error-correcting codes
based on algebraic curves over finite fields, and list decoding
algorithms for them. The basic goal in the subject of list decoding is
to construct error-correcting codes $C$ over some alphabet $\Sigma$
which have good rate $R$, and at ... more >>>

TR05-133 | 17th November 2005
Venkatesan Guruswami, Atri Rudra

Explicit Capacity-Achieving List-Decodable Codes

Revisions: 1

For every $0 < R < 1$ and $\eps > 0$, we present an explicit
construction of error-correcting codes of rate $R$ that can be list
decoded in polynomial time up to a fraction $(1-R-\eps)$ of errors.
These codes achieve the ``capacity'' for decoding from {\em ... more >>>

TR05-134 | 17th November 2005
Xi Chen, Xiaotie Deng

3-NASH is PPAD-Complete

In this paper, we improve a recent result of Daskalakis, Goldberg and Papadimitriou on PPAD-completeness of 4-Nash, showing that 3-Nash is PPAD-complete.

more >>>

TR05-135 | 19th November 2005
Iftach Haitner, Danny Harnik, Omer Reingold

On the Power of the Randomized Iterate

We consider two of the most fundamental theorems in Cryptography. The first, due to Haastad et. al. [HILL99], is that pseudorandom generators can be constructed from any one-way function. The second due to Yao [Yao82] states that the existence of weak one-way functions (i.e. functions on which every efficient algorithm ... more >>>

TR05-136 | 14th November 2005
Anna Gal, Michal Koucky, Pierre McKenzie

Incremental branching programs

In this paper we propose the study of a new model of restricted
branching programs which we call incremental branching programs.
This is in line with the program proposed by Cook in 1974 as an
approach for separating the class of problems solvable in logarithmic
space from problems solvable ... more >>>

TR05-137 | 21st November 2005
Emanuele Viola

On Probabilistic Time versus Alternating Time

We prove several new results regarding the relationship between probabilistic time, BPTime(t), and alternating time, \Sigma_{O(1)} Time(t). Our main results are the following:

1) We prove that BPTime(t) \subseteq \Sigma_3 Time(t polylog(t)). Previous results show that BPTime(t) \subseteq \Sigma_2 Time(t^2 log t) (Sipser and Gacs, STOC '83; Lautemann, IPL '83) ... more >>>

TR05-138 | 22nd November 2005
Peter Bürgisser, Felipe Cucker

Exotic quantifiers, complexity classes, and complete problems

We introduce some operators defining new complexity classes from existing ones in the Blum-Shub-Smale theory of computation over the reals. Each one of these operators is defined with the help of a quantifier differing from the usual ones, $\forall$ and $\exists$, and yet having a precise geometric meaning. Our agenda ... more >>>

TR05-139 | 21st November 2005
Constantinos Daskalakis, Christos H. Papadimitriou

Three-Player Games Are Hard

We prove that computing a Nash equilibrium in a 3-player
game is PPAD-complete, solving a problem left open in our recent result on the complexity of Nash equilibria.

more >>>

TR05-140 | 30th November 2005
Xi Chen, Xiaotie Deng

Settling the Complexity of 2-Player Nash-Equilibrium

Revisions: 1

We prove that finding the solution of two player Nash Equilibrium is PPAD-complete.

more >>>

TR05-141 | 29th November 2005
Amos Beimel, Paz Carmi, Kobbi Nissim, Enav Weinreb

Private Approximation of Search Problems

Many approximation algorithms have been presented in the last decades
for hard search problems. The focus of this paper is on cryptographic
applications, where it is desired to design algorithms which do not
leak unnecessary information. Specifically, we are interested in
private approximation algorithms -- efficient algorithms ... more >>>

TR05-142 | 1st December 2005
Vadim Lyubashevsky, Daniele Micciancio

Generalized Compact Knapsacks are Collision Resistant

The generalized knapsack problem is the following: given $m$ random
elements $a_1,\ldots,a_m\in R$ for some ring $R$, and a target $t\in
R$, find elements $z_1,\ldots,z_m\in D$ such that $\sum{a_iz_i}=t$
where $D$ is some given subset of $R$. In (Micciancio, FOCS 2002),
it was proved that for appropriate choices of $R$ ... more >>>

TR05-143 | 29th November 2005
Parikshit Gopalan

Constructing Ramsey Graphs from Boolean Function Representations

Explicit construction of Ramsey graphs or graphs with no large clique or independent set has remained a challenging open problem for a long time. While Erdos's probabilistic argument shows the existence of graphs on 2^n vertices with no clique or independent set of size 2n, the best known explicit constructions ... more >>>

TR05-144 | 5th December 2005
Lance Fortnow, Luis Antunes

Time-Bounded Universal Distributions

We show that under a reasonable hardness assumptions, the time-bounded Kolmogorov distribution is a universal samplable distribution. Under the same assumption we exactly characterize the worst-case running time of languages that are in average polynomial-time over all P-samplable distributions.

more >>>

TR05-145 | 5th December 2005
Ronen Shaltiel

How to get more mileage from randomness extractors

Let $\cal C$ be a class of distributions over $\B^n$. A deterministic randomness extractor for $\cal C$ is a function $E:\B^n \ar \B^m$ such that for any $X$ in $\cal C$ the distribution $E(X)$ is statistically close to the uniform distribution. A long line of research deals with explicit constructions ... more >>>

TR05-146 | 25th November 2005
Gábor Erdèlyi, Tobias Riege, Jörg Rothe

Quantum Cryptography: A Survey

Revisions: 2

We survey some results in quantum cryptography. After a brief
introduction to classical cryptography, we provide the physical and
mathematical background needed and present some fundamental protocols
from quantum cryptography, including quantum key distribution and
quantum bit commitment protocols.

more >>>

TR05-147 | 5th December 2005
Christian Glaßer, Stephen Travers

Machines that can Output Empty Words

We propose the e-model for leaf languages which generalizes the known balanced and unbalanced concepts. Inspired by the neutral behavior of rejecting paths of NP machines, we allow transducers to output empty words.

The paper explains several advantages of the new model. A central aspect is that it allows us ... more >>>

TR05-148 | 6th December 2005
Eric Allender, Samir Datta, Sambuddha Roy

The Directed Planar Reachability Problem

Revisions: 1

We investigate the s-t-connectivity problem for directed planar graphs, which is hard for L and is contained in NL but is not known to be complete. We show that this problem is logspace-reducible to its complement, and we show that the problem of searching graphs of genus 1 reduces to ... more >>>

TR05-149 | 7th December 2005
Eric Allender, David Mix Barrington, Tanmoy Chakraborty, Samir Datta, Sambuddha Roy

Grid Graph Reachability Problems

Revisions: 1

We study the complexity of restricted versions of st-connectivity, which is the standard complete problem for NL. Grid graphs are a useful tool in this regard, since
* reachability on grid graphs is logspace-equivalent to reachability in general planar digraphs, and
* reachability on certain classes of grid graphs gives ... more >>>

TR05-150 | 5th December 2005
Neeraj Kayal, Nitin Saxena

Polynomial Identity Testing for Depth 3 Circuits

We study the identity testing problem for depth $3$ arithmetic circuits ($\Sigma\Pi\Sigma$ circuits). We give the first deterministic polynomial time identity test for $\Sigma\Pi\Sigma$ circuits with bounded top fanin. We also show that the {\em rank} of a minimal and simple $\Sigma\Pi\Sigma$ circuit with bounded top fanin, computing zero, can ... more >>>

TR05-151 | 7th December 2005
Magnus Bordewich, Martin Dyer, Marek Karpinski

Metric Construction, Stopping Times and Path Coupling.

In this paper we examine the importance of the choice of metric in path coupling, and the relationship of this to \emph{stopping time analysis}. We give strong evidence that stopping time analysis is no more powerful than standard path coupling. In particular, we prove a stronger theorem for path coupling ... more >>>

TR05-152 | 9th December 2005
Oded Lachish, Ilan Newman

Languages that are Recognized by Simple Counter Automata are not necessarily Testable

Combinatorial property testing deals with the following relaxation of
decision problems: Given a fixed property and an input $f$, one wants
to decide whether $f$ satisfies the property or is `far' from satisfying
the property.
It has been shown that regular languages are testable,
and that there exist context free ... more >>>

TR05-153 | 9th December 2005
Shirley Halevy, Oded Lachish, Ilan Newman, Dekel Tsur

Testing Orientation Properties

We propose a new model for studying graph related problems
that we call the \emph{orientation model}. In this model, an undirected
graph $G$ is fixed, and the input is any possible edge orientation
of $G$. A property is now a property of the directed graph that is
obtained by a ... more >>>

TR05-154 | 11th December 2005
Albert Atserias

Non-Uniform Hardness for NP via Black-Box Adversaries

We may believe SAT does not have small Boolean circuits.
But is it possible that some language with small circuits
looks indistiguishable from SAT to every polynomial-time
bounded adversary? We rule out this possibility. More
precisely, assuming SAT does not have small circuits, we
show that ... more >>>

TR05-155 | 10th December 2005
Amir Shpilka

Constructions of low-degree and error-correcting epsilon-biased sets

In this work we give two new constructions of $\epsilon$-biased
generators. Our first construction answers an open question of
Dodis and Smith, and our second construction
significantly extends a result of Mossel et al.
In particular we obtain the following results:

1. We construct a family of asymptotically good binary ... more >>>

TR05-156 | 13th December 2005
Jonathan A. Kelner, Daniel A. Spielman

A Randomized Polynomial-Time Simplex Algorithm for Linear Programming (Preliminary Version)

We present the first randomized polynomial-time simplex algorithm for linear programming. Like the other known polynomial-time algorithms for linear programming, its running time depends polynomially on the number of bits used to represent its input.

We begin by reducing the input linear program to a special form in which we ... more >>>

TR05-157 | 10th December 2005
Xiaoyang Gu, Jack H. Lutz, Elvira Mayordomo

Points on Computable Curves

The ``analyst's traveling salesman theorem'' of geometric
measure theory characterizes those subsets of Euclidean
space that are contained in curves of finite length.
This result, proven for the plane by Jones (1990) and
extended to higher-dimensional Euclidean spaces by
Okikiolu (1991), says that a bounded set $K$ is contained
more >>>

TR05-158 | 12th December 2005
Chris Peikert, Alon Rosen

Efficient Collision-Resistant Hashing from Worst-Case Assumptions on Cyclic Lattices

The generalized knapsack function is defined as $f_{\a}(\x) = \sum_i
a_i \cdot x_i$, where $\a = (a_1, \ldots, a_m)$ consists of $m$
elements from some ring $R$, and $\x = (x_1, \ldots, x_m)$ consists
of $m$ coefficients from a specified subset $S \subseteq R$.
Micciancio ... more >>>

TR05-159 | 14th November 2005
Daniel Rolf

Improved Bound for the PPSZ/Schöning-Algorithm for $3$-SAT

The PPSZ Algorithm presented by Paturi, Pudlak, Saks, and Zane in 1998 has the nice feature that the only satisfying solution of a uniquely satisfiable $3$-SAT formula can be found in expected running time at most $O(1.3071^n)$. Its bound degenerates when the number of solutions increases. In 1999, Schöning proved ... more >>>

TR05-160 | 10th December 2005
Xiaoyang Gu, Jack H. Lutz

Dimension Characterizations of Complexity Classes

We use derandomization to show that sequences of positive $\pspace$-dimension -- in fact, even positive $\Delta^\p_k$-dimension
for suitable $k$ -- have, for many purposes, the full power of random oracles. For example, we show that, if $S$ is any binary sequence whose $\Delta^p_3$-dimension is positive, then $\BPP\subseteq \P^S$ and, moreover, ... more >>>

TR05-161 | 13th December 2005
John Hitchcock

Online Learning and Resource-Bounded Dimension: Winnow Yields New Lower Bounds for Hard Sets

We establish a relationship between the online mistake-bound model of learning and resource-bounded dimension. This connection is combined with the Winnow algorithm to obtain new results about the density of hard sets under adaptive reductions. This improves previous work of Fu (1995) and Lutz and Zhao (2000), and solves one ... more >>>

TR05-162 | 23rd December 2005
Yunlei Zhao, Jesper Buus Nielsen, Robert H. Deng, Feng Dengguo

Generic yet Practical ZK Arguments from any Public-Coin HVZK

Revisions: 1

In this work, we present a generic yet practical transformation from any public-coin honest-verifier zero-knowledge (HVZK) protocols to normal zero-knowledge (ZK) arguments. By ``generic", we mean that the transformation is applicable to any public-coin HVZK protocol under any one-way function (OWF) admitting Sigma-protocols. By ``practical" we mean that the transformation ... more >>>

TR05-163 | 19th December 2005
Dvir Falik, Alex Samorodnitsky

Edge-isoperimetric inequalities and influences

We give a combinatorial proof of the result of Kahn, Kalai, and Linial, which states that every balanced boolean function on the n-dimensional boolean cube has a variable with influence of at least Omega(log(n)/n).The methods of the proof are then used to recover additional isoperimetric results for the cube, with ... more >>>

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