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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 >>>

TR05-047 | 10th April 2005

#### 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 >>>

TR11-072 | 1st May 2011
Danny Hermelin, Xi Wu

#### Weak Compositions and Their Applications to Polynomial Lower-Bounds for Kernelization

Revisions: 1

We introduce a new form of composition called \emph{weak composition} that allows us to obtain polynomial kernelization lower-bounds for several natural parameterized problems. Let $d \ge 2$ be some constant and let $L_1, L_2 \subseteq \{0,1\}^* \times \N$ be two parameterized problems where the unparameterized version of $L_1$ is \NP-hard. ... more >>>

TR10-005 | 3rd January 2010
Haitao Jiang, Binhai Zhu

#### Weak Kernels

Revisions: 7

In this paper, we formalize a folklore concept and formally define
{\em weak kernels} for fixed-parameter computation. We show that a
problem has a (traditional) kernel then it also has a weak kernel.
It is unknown yet whether the converse is always true. On the other hand,
for a problem ... more >>>

TR13-164 | 28th November 2013
Scott Aaronson, Andris Ambainis, Kaspars Balodis, Mohammad Bavarian

#### Weak Parity

We study the query complexity of Weak Parity: the problem of computing the parity of an n-bit input string, where one only has to succeed on a 1/2+eps fraction of input strings, but must do so with high probability on those inputs where one does succeed. It is well-known that ... more >>>

TR97-011 | 7th April 1997
Alexander E. Andreev, Andrea E. F. Clementi, Jose' D.P. Rolim and Trevisan

#### Weak Random Sources, Hitting Sets, and BPP Simulations

We show how to simulate any BPP algorithm in polynomial time
using a weak random source of min-entropy $r^{\gamma}$
for any $\gamma >0$.
This follows from a more general result about {\em sampling\/}
with weak random sources.
Our result matches an information-theoretic lower bound ... more >>>

TR15-170 | 26th October 2015
Alexander Golovnev, Alexander Kulikov

#### Weighted gate elimination: Boolean dispersers for quadratic varieties imply improved circuit lower bounds

In this paper we motivate the study of Boolean dispersers for quadratic varieties by showing that an explicit construction of such objects gives improved circuit lower bounds. An $(n,k,s)$-quadratic disperser is a function on $n$ variables that is not constant on any subset of $\mathbb{F}_2^n$ of size at least $s$ ... more >>>

TR14-166 | 8th December 2014
Mark Bun, Thomas Steinke

#### Weighted Polynomial Approximations: Limits for Learning and Pseudorandomness

Polynomial approximations to boolean functions have led to many positive results in computer science. In particular, polynomial approximations to the sign function underly algorithms for agnostically learning halfspaces, as well as pseudorandom generators for halfspaces. In this work, we investigate the limits of these techniques by proving inapproximability results for ... more >>>

TR17-083 | 5th May 2017

#### Weights at the Bottom Matter When the Top is Heavy

Revisions: 1

Proving super-polynomial lower bounds against depth-2 threshold circuits of the form THR of THR is a well-known open problem that represents a frontier of our understanding in boolean circuit complexity. By contrast, exponential lower bounds on the size of THR of MAJ circuits were shown by Razborov and Sherstov (SIAM ... more >>>

TR13-095 | 24th June 2013
Uriel Feige, Rani Izsak

#### Welfare Maximization and the Supermodular Degree

Given a set of items and a collection of players, each with a nonnegative monotone valuation set function over the items,
the welfare maximization problem requires that every item be allocated to exactly one player,
and one wishes to maximize the sum of values obtained by the players,
as computed ... more >>>

TR15-054 | 7th April 2015
Noga Alon, Noam Nisan, Ran Raz, Omri Weinstein

#### Welfare Maximization with Limited Interaction

We continue the study of welfare maximization in unit-demand (matching) markets, in a distributed information model
where agent's valuations are unknown to the central planner, and therefore communication is required to determine an
efficient allocation. Dobzinski, Nisan and Oren (STOC'14) showed that if the market size is $n$, ... more >>>

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

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

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

TR01-084 | 1st October 2001
Gerhard J. Woeginger

#### When does a dynamic programming formulation guarantee the existence of an FPTAS?

We derive results of the following flavor:
If a combinatorial optimization problem can be formulated via a dynamic
program of a certain structure and if the involved cost and transition
functions satisfy certain arithmetical and structural conditions, then
the optimization problem automatically possesses a fully polynomial time
approximation scheme (FPTAS).

... more >>>

TR06-065 | 24th May 2006
Jan Arpe, Rüdiger Reischuk

#### When Does Greedy Learning of Relevant Features Succeed? --- A Fourier-based Characterization ---

Detecting the relevant attributes of an unknown target concept
is an important and well studied problem in algorithmic learning.
Simple greedy strategies have been proposed that seem to perform reasonably
well in practice if a sufficiently large random subset of examples of the target
concept is provided.

Introducing a ... more >>>

TR18-002 | 31st December 2017
Constantinos Daskalakis, Gautam Kamath, John Wright

#### Which Distribution Distances are Sublinearly Testable?

Given samples from an unknown distribution $p$ and a description of a distribution $q$, are $p$ and $q$ close or far? This question of "identity testing" has received significant attention in the case of testing whether $p$ and $q$ are equal or far in total variation distance. However, in recent ... more >>>

TR07-065 | 13th July 2007
Jonathan Katz

#### Which Languages Have 4-Round Zero-Knowledge Proofs?

We show that if a language $L$ has a 4-round, black-box, computational zero-knowledge proof system with negligible soundness error, then $\bar L \in MA$. Assuming the polynomial hierarchy does not collapse, this means, in particular, that $NP$-complete languages do not have 4-round zero-knowledge proofs (at least with respect to black-box ... more >>>

TR17-015 | 4th February 2017
Ilan Komargodski, Moni Naor, Eylon Yogev

#### White-Box vs. Black-Box Complexity of Search Problems: Ramsey and Graph Property Testing

Revisions: 1

Ramsey theory assures us that in any graph there is a clique or independent set of a certain size, roughly logarithmic in the graph size. But how difficult is it to find the clique or independent set? If the graph is given explicitly, then it is possible to do so ... more >>>

TR11-108 | 8th August 2011
Scott Aaronson

#### Why Philosophers Should Care About Computational Complexity

Revisions: 2

One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed case that one would be wrong. In particular, I argue that computational complexity theory---the field that studies the ... more >>>

TR15-159 | 28th September 2015
Juraj Hromkovic

#### Why the Concept of Computational Complexity is Hard for Verifiable Mathematics

Mathematics was developed as a strong research instrument with fully verifiable argumentations. We call any formal theory based on syntactic rules that enables to algorithmically verify for any given text whether it is a proof or not algorithmically verifiable mathematics (AV-mathematics for short). We say that a decision problem L ... more >>>

TR15-048 | 14th February 2015

#### Width Hierarchy for $k$-OBDD of Small Width

Revisions: 1

In this paper was explored well known model $k$-OBDD. There are proven width based hierarchy of classes of boolean functions which computed by $k$-OBDD. The proof of hierarchy is based on sufficient condition of Boolean function's non representation as $k$-OBDD and complexity properties of Boolean
function SAF. This function is ... more >>>

TR06-092 | 5th July 2006
Matthias Englert, Heiko Röglin, Berthold Vöcking

#### Worst Case and Probabilistic Analysis of the 2-Opt Algorithm for the TSP

2-Opt is probably the most basic and widely used local search
heuristic for the TSP. This heuristic achieves amazingly good
results on "real world" Euclidean instances both with respect to
running time and approximation ratio. There are numerous
experimental studies on the performance of 2-Opt. However, the
theoretical knowledge about ... more >>>

TR08-072 | 11th August 2008
Shachar Lovett, Tali Kaufman

#### Worst case to Average case reductions for polynomials

A degree-d polynomial p in n variables over a field F is equidistributed if it takes on each of its |F| values close to equally often, and biased otherwise. We say that p has low rank if it can be expressed as a function of a small number of lower ... more >>>

TR18-056 | 20th March 2018
Zvika Brakerski, Vadim Lyubashevsky, Vinod Vaikuntanathan, Daniel Wichs

#### Worst-Case Hardness for LPN and Cryptographic Hashing via Code Smoothing

We present a worst case decoding problem whose hardness reduces to that of solving the Learning Parity with Noise (LPN) problem, in some parameter regime. Prior to this work, no worst case hardness result was known for LPN (as opposed to syntactically similar problems such as Learning with Errors). The ... more >>>

TR00-019 | 20th March 2000
Edward Hirsch

#### Worst-case time bounds for MAX-k-SAT w.r.t. the number of variables using local search

During the past three years there was an explosion of algorithms
solving MAX-SAT and MAX-2-SAT in worst-case time of the order
c^K, where c<2 is a constant, and K is the number of clauses
in the input formula. Such bounds w.r.t. the number of variables
instead of the number of ... more >>>

TR18-090 | 4th May 2018
Eli Ben-Sasson, Swastik Kopparty, Shubhangi Saraf

#### Worst-case to average case reductions for the distance to a code

Revisions: 1

Algebraic proof systems reduce computational problems to problems about estimating the distance of a sequence of functions $u=(u_1,\ldots, u_k)$, given as oracles, from a linear error correcting code $V$. The soundness of such systems relies on methods that act locally'' on $u$ and map it to a single function $u^*$ ... more >>>

TR17-130 | 30th August 2017
Oded Goldreich, Guy Rothblum

#### Worst-case to Average-case reductions for subclasses of P

Revisions: 4

For every polynomial $q$, we present worst-case to average-case (almost-linear-time) reductions for a class of problems in $\cal P$ that are widely conjectured not to be solvable in time $q$.
These classes contain, for example, the problems of counting the number of $k$-cliques in a graph, for any fixed $k\geq3$.
more >>>

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