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Electronic Colloquium on Computational Complexity

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All reports by Author Johan Hastad:

TR16-041 | 17th March 2016
Johan Hastad

An average-case depth hierarchy theorem for higher depth

We extend the recent hierarchy results of Rossman, Servedio and
Tan \cite{rst15} to any $d \leq \frac {c \log n}{\log {\log n}}$
for an explicit constant $c$.

To be more precise, we prove that for any such $d$ there is a function
$F_d$ that is computable by a read-once formula ... more >>>

TR12-137 | 1st November 2012
Johan Hastad

On the correlation of parity and small-depth circuits

Revisions: 1

We prove that the correlation of a depth-$d$
unbounded fanin circuit of size $S$ with parity
of $n$ variables is at most $2^{-\Omega(n/(\log S)^{d-1})}$.

more >>>

TR11-142 | 2nd November 2011
Boaz Barak, Parikshit Gopalan, Johan Hastad, Raghu Meka, Prasad Raghavendra, David Steurer

Making the long code shorter, with applications to the Unique Games Conjecture

Revisions: 1

The long code is a central tool in hardness of approximation, especially in
questions related to the unique games conjecture. We construct a new code that
is exponentially more ecient, but can still be used in many of these applications.
Using the new code we obtain exponential improvements over several ... more >>>

TR11-027 | 28th February 2011
Venkatesan Guruswami, Johan Hastad, Rajsekar Manokaran, Prasad Raghavendra, Moses Charikar

Beating the Random Ordering is Hard: Every ordering CSP is approximation resistant

We prove that, assuming the Unique Games Conjecture (UGC), every problem in the class of ordering constraint satisfaction problems (OCSP) where each constraint has constant arity is approximation
resistant. In other words, we show that if $\rho$ is the expected fraction of constraints satisfied by a random ordering, then obtaining ... more >>>

TR10-132 | 18th August 2010
Mahdi Cheraghchi, Johan Hastad, Marcus Isaksson, Ola Svensson

Approximating Linear Threshold Predicates

We study constraint satisfaction problems on the domain $\{-1,1\}$, where the given constraints are homogeneous linear threshold predicates. That is, predicates of the form $\mathrm{sgn}(w_1 x_1 + \cdots + w_n x_n)$ for some positive integer weights $w_1, \dots, w_n$. Despite their simplicity, current techniques fall short of providing a classification ... more >>>

TR10-003 | 6th January 2010
Venkatesan Guruswami, Johan Hastad, Swastik Kopparty

On the List-Decodability of Random Linear Codes

For every fixed finite field $\F_q$, $p \in (0,1-1/q)$ and $\varepsilon >
0$, we prove that with high probability a random subspace $C$ of
$\F_q^n$ of dimension $(1-H_q(p)-\varepsilon)n$ has the
property that every Hamming ball of radius $pn$ has at most
$O(1/\varepsilon)$ codewords.

This ... more >>>

TR08-026 | 28th February 2008
Jakob Nordström, Johan Hastad

Towards an Optimal Separation of Space and Length in Resolution

Most state-of-the-art satisfiability algorithms today are variants of
the DPLL procedure augmented with clause learning. The main bottleneck
for such algorithms, other than the obvious one of time, is the amount
of memory used. In the field of proof complexity, the resources of
time and memory correspond to the length ... more >>>

TR00-062 | 25th August 2000
Venkatesan Guruswami, Johan Hastad, Madhu Sudan

Hardness of approximate hypergraph coloring

We introduce the notion of covering complexity of a probabilistic
verifier. The covering complexity of a verifier on a given input is
the minimum number of proofs needed to ``satisfy'' the verifier on
every random string, i.e., on every random string, at least one of the
given proofs must be ... more >>>

TR99-039 | 24th September 1999
Johan Hastad

On approximating CSP-B

We prove that any constraint satisfaction problem
where each variable appears a bounded number of
times admits a nontrivial polynomial time approximation

more >>>

TR99-037 | 27th August 1999
Johan Hastad, Mats Näslund

The Security of all RSA and Discrete Log Bits

We study the security of individual bits in an
RSA encrypted message $E_N(x)$. We show that given $E_N(x)$,
predicting any single bit in $x$ with only a non-negligible
advantage over the trivial guessing strategy, is (through a
polynomial time reduction) as hard as breaking ... more >>>

TR99-025 | 2nd July 1999
Yonatan Aumann, Johan Hastad, Michael O. Rabin, Madhu Sudan

Linear Consistency Testing

We extend the notion of linearity testing to the task of checking
linear-consistency of multiple functions. Informally, functions
are ``linear'' if their graphs form straight lines on the plane.
Two such functions are ``consistent'' if the lines have the same
slope. We propose a variant of a test of ... more >>>

TR96-018 | 23rd February 1996
Oded Goldreich, Johan Hastad

On the Message Complexity of Interactive Proof Systems

Revisions: 2

We investigate the computational complexity of languages
which have interactive proof systems of bounded message complexity.
In particular, we show that
(1) If $L$ has an interactive proof in which the total
communication is bounded by $c(n)$ bits
then $L$ can be recognized a probabilitic machine
in time ... more >>>

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