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Electronic Colloquium on Computational Complexity

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Reports tagged with complexity:
TR95-014 | 27th January 1995
U. Faigle, W. Kern, M. Streng

Note On the Computational Complexity of $j$-Radii of Polytopes in ${\Re}^n$

We show that, for fixed dimension $n$, the approximation of
inner and outer $j$-radii of polytopes in ${\Re}^n$, endowed
with the Euclidean norm, is polynomial.

more >>>

TR95-038 | 2nd July 1995
Joe Kilian, Erez Petrank

An Efficient Non-Interactive Zero-Knowledge Proof System for NP with General Assumptions

We consider noninteractive zero-knowledge proofs in the shared random
string model proposed by Blum, Feldman and Micali \cite{bfm}. Until
recently there was a sizable polynomial gap between the most
efficient noninteractive proofs for {\sf NP} based on general
complexity assumptions \cite{fls} versus those based on specific
algebraic assumptions \cite{Da}. ... more >>>

TR96-036 | 28th May 1996
Petr Savicky, Stanislav Zak

A large lower bound for 1-branching programs

Revisions: 1

Branching programs (b.p.'s) or decision diagrams are a general
graph-based model of sequential computation. B.p.'s of polynomial
size are a nonuniform counterpart of LOG. Lower bounds for
different kinds of restricted b.p.'s are intensively investigated.
An important restriction are so called 1-b.p.'s, where each
computation reads each input bit at ... more >>>

TR97-021 | 16th May 1997
Farid Ablayev

Randomization and nondeterminsm are incomparable for ordered read-once branching programs

In the manuscript F. Ablayev and M. Karpinski, On the power of
randomized branching programs (generalization of ICALP'96 paper
results for the case of pure boolean function, available at
http://www.ksu.ru/~ablayev) we exhibited a simple boolean functions
$f_n$ in $n$ variables such that:

1) $f_{n}$ can be computed ... more >>>

TR97-046 | 3rd October 1997
Alexander Barg

Complexity Issues in Coding Theory

This is a research-expository paper. It deals with
complexity issues in the theory of linear block codes. The main
emphasis is on the theoretical performance limits of the
best known codes. Therefore, the main subject of the paper are
families of asymptotically good codes, i.e., codes whose rate and
relative ... more >>>

TR97-057 | 3rd November 1997
Petr Savicky

Complexity and Probability of some Boolean Formulas

For any Boolean function $f$ let $L(f)$ be its formula size
complexity in the basis $\{\land,\oplus,1\}$. For every $n$ and
every $k\le n/2$, we describe a probabilistic distribution
on formulas in the basis $\{\land,\oplus,1\}$ in some given set of
$n$ variables and of the ... more >>>

TR98-008 | 15th January 1998
Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, Mario Szegedy

Proof verification and the hardness of approximation problems.

We show that every language in NP has a probablistic verifier
that checks membership proofs for it using
logarithmic number of random bits and by examining a
<em> constant </em> number of bits in the proof.
If a string is in the language, then there exists a proof ... more >>>

TR98-034 | 23rd June 1998
Venkatesan Guruswami, Daniel Lewin and Madhu Sudan, Luca Trevisan

A tight characterization of NP with 3 query PCPs

It is known that there exists a PCP characterization of NP
where the verifier makes 3 queries and has a {\em one-sided}
error that is bounded away from 1; and also that 2 queries
do not suffice for such a characterization. Thus PCPs with
3 ... more >>>

TR98-040 | 24th July 1998
Madhu Sudan, Luca Trevisan

Probabilistically checkable proofs with low amortized query complexity

The error probability of Probabilistically Checkable Proof (PCP)
systems can be made exponentially small in the number of queries
by using sequential repetition. In this paper we are interested
in determining the precise rate at which the error goes down in
an optimal protocol, and we make substantial progress toward ... more >>>

TR99-022 | 14th June 1999
Eli Ben-Sasson, Avi Wigderson

Short Proofs are Narrow - Resolution made Simple

The width of a Resolution proof is defined to be the maximal number of
literals in any clause of the proof. In this paper we relate proof width
to proof length (=size), in both general Resolution, and its tree-like
variant. The following consequences of these relations reveal width as ... more >>>

TR99-047 | 10th November 1999
Wolfgang Slany

Graph Ramsey games

We consider combinatorial avoidance and achievement games
based on graph Ramsey theory: The players take turns in coloring
still uncolored edges of a graph G, each player being assigned a
distinct color, choosing one edge per move. In avoidance games,
completing a monochromatic subgraph isomorphic to ... more >>>

TR00-006 | 26th January 2000
E.A. Okol'nishnikiva

On operations of geometrical projection and monotone extension

Some operations over Boolean functions are considered. It is shown that
the operation of the geometrical projection and the operation of the
monotone extension can increase the complexity of Boolean functions for
formulas in each finite basis, for switching networks, for branching
programs, and read-$k$-times ... more >>>

TR01-079 | 6th September 2001
Michele Zito

An Upper Bound on the Space Complexity of Random Formulae in Resolution

We prove that, with high probability, the space complexity of refuting
a random unsatisfiable boolean formula in $k$-CNF on $n$
variables and $m = \Delta n$ clauses is
$O(n \cdot \Delta^{-\frac{1}{k-2}})$.

more >>>

TR02-032 | 17th April 2002
Andrei Bulatov

Tractable Constraint Satisfaction Problems on a 3-element set

The Constraint Satisfaction Problem (CSP) provides a common framework for many combinatorial problems. The general CSP is known to be NP-complete; however, certain restrictions on a possible form of constraints may affect the complexity, and lead to tractable problem classes. There is, therefore, a fundamental research direction, aiming to separate ... more >>>

TR02-034 | 18th April 2002
Andrei Bulatov

Mal'tsev constraints are tractable

A wide variety of combinatorial problems can be represented in the form of the Constraint Satisfaction Problem (CSP). The general CSR is known to be NP-complete, however, some restrictions on the possible form of constraints may lead to a tractable subclass. Jeavons and coauthors have shown that the complexity of ... 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 >>>

TR03-047 | 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 >>>

TR03-053 | 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 >>>

TR03-062 | 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 >>>

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

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

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

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 >>>

TR05-121 | 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 >>>

TR06-051 | 8th April 2006
Alan Nash, Russell Impagliazzo, Jeff Remmel

Infinitely-Often Universal Languages and Diagonalization

Diagonalization is a powerful technique in recursion theory and in
computational complexity \cite{For00}. The limits of this technique are
not clear. On the one hand, many people argue that conflicting
relativizations mean a complexity question cannot be resolved using only
diagonalization. On the other hand, it is not clear that ... more >>>

TR06-094 | 29th July 2006
Parikshit Gopalan, Phokion G. Kolaitis, Elitza Maneva, Christos H. Papadimitriou

The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies

Revisions: 1

Boolean satisfiability problems are an important benchmark for questions about complexity, algorithms, heuristics and threshold phenomena. Recent work on heuristics, and the satisfiability threshold has centered around the structure and connectivity of the solution space. Motivated by this work, we study structural and connectivity-related properties of the space of solutions ... more >>>

TR06-103 | 5th July 2006
Oded Lachish, Ilan Newman, Asaf Shapira

Space Complexity vs. Query Complexity

Combinatorial property testing deals with the following relaxation
of decision problems: Given a fixed property and an input $x$, one
wants to decide whether $x$ satisfies the property or is ``far''
from satisfying it. The main focus of property testing is in
identifying large families of properties that can be ... more >>>

TR06-120 | 12th September 2006
Leslie G. Valiant


Living cells function according to complex mechanisms that operate in different ways depending on conditions. Evolutionary theory suggests that such mechanisms evolved as a result of a random search guided by selection and realized by genetic mutations. However, as some observers have noted, there has existed no theory that would ... more >>>

TR07-004 | 12th January 2007
Lance Fortnow, Rahul Santhanam

Time Hierarchies: A Survey

We survey time hierarchies, with an emphasis on recent attempts to prove hierarchies for semantic classes.

more >>>

TR07-093 | 27th July 2007
Andrei A. Bulatov

The complexity of the counting constraint satisfaction problem

Revisions: 1

The Counting Constraint Satisfaction Problem (#CSP(H)) over a finite
relational structure H can be expressed as follows: given a
relational structure G over the same vocabulary,
determine the number of homomorphisms from G to H.
In this paper we characterize relational structures H for which
#CSP(H) can be solved in ... more >>>

TR08-032 | 18th March 2008
Dmitriy Cherukhin

Lower Bounds for Boolean Circuits with Finite Depth and Arbitrary Gates

We consider bounded depth circuits over an arbitrary field $K$. If the field $K$ is finite, then we allow arbitrary gates $K^n\to K$. For instance, in the case of field $GF(2)$ we allow any Boolean gates. If the field $K$ is infinite, then we allow only polinomials.

For every fixed ... more >>>

TR08-050 | 12th March 2008
Manoj Prabhakaran, Mike Rosulek

Cryptographic Complexity of Multi-party Computation Problems: Classifications and Separations

We develop new tools to study the relative complexities of secure
multi-party computation tasks (functionalities) in the Universal
Composition framework. When one task can be securely realized using
another task as a black-box, we interpret this as a
qualitative, complexity-theoretic reduction between the two tasks.
Virtually all previous characterizations of ... more >>>

TR09-025 | 11th March 2009
Arnaldo Moura, Igor Carboni Oliveira

A New Look at Some Classical Results in Computational Complexity

We propose a generalization of the traditional algorithmic space and
time complexities. Using the concept introduced, we derive an
unified proof for the deterministic time and space hierarchy
theorems, now stated in a much more general setting. This opens the
possibility for the unification and generalization of other results
that ... more >>>

TR09-035 | 26th March 2009
Nicola Galesi, Massimo Lauria

On the Automatizability of Polynomial Calculus

We prove that Polynomial Calculus and Polynomial Calculus with Resolution are not automatizable, unless W[P]-hard problems are fixed parameter tractable by one-side error randomized algorithms. This extends to Polynomial Calculus the analogous result obtained for Resolution by Alekhnovich and Razborov (SIAM J. Computing, 38(4), 2008).

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TR09-068 | 1st September 2009
Dave Buchfuhrer, Chris Umans

Limits on the Social Welfare of Maximal-In-Range Auction Mechanisms

Many commonly-used auction mechanisms are ``maximal-in-range''. We show that any maximal-in-range mechanism for $n$ bidders and $m$ items cannot both approximate the social welfare with a ratio better than $\min(n, m^\eta)$ for any constant $\eta < 1/2$ and run in polynomial time, unless $NP \subseteq P/poly$. This significantly improves upon ... more >>>

TR10-154 | 8th October 2010
Derrick Stolee, Vinodchandran Variyam

Space-Efficient Algorithms for Reachability in Surface-Embedded Graphs

We consider the reachability problem for a certain class of directed acyclic graphs embedded on surfaces. Let ${\cal G}(m,g)$ be the class of directed acyclic graphs with $m = m(n)$ source vertices embedded on a surface (orientable or non-orientable) of genus $g = g(n)$. We give a log-space reduction that ... more >>>

TR11-130 | 25th September 2011
Sergei Lozhkin, Alexander Shiganov

On a Modification of Lupanov's Method with More Uniform Distribution of Fan-out

In this paper we suggest a modification of classical Lupanov's method [Lupanov1958]
that allows building circuits over the basis $\{\&,\vee,\neg\}$ for Boolean functions of $n$ variables with size at most
\frac{2^n}{n}\left(1+\frac{3\log n + O(1)}{n}\right),
and with more uniform distribution of outgoing arcs by circuit gates.

For almost all ... more >>>

TR13-019 | 31st January 2013
Stephen A. Fenner, Rohit Gurjar, Arpita Korwar, Thomas Thierauf

On Two-Level Poset Games

We consider the complexity of determining the winner of a finite, two-level poset game.
This is a natural question, as it has been shown recently that determining the winner of a finite, three-level poset game is PSPACE-complete.
We give a simple formula allowing one to compute the status ... more >>>

TR13-041 | 14th March 2013
Igor Sergeev

On the complexity of parallel prefix circuits

It is shown that complexity of implementation of prefix sums of $m$ variables (i.e. functions $x_1 \cdot \ldots\cdot x_i$, $1\le i \le m$) by circuits of depth $\lceil \log_2 m \rceil$ in the case $m=2^n$ is exactly $$3.5\cdot2^n - (8.5+3.5(n \bmod 2))2^{\lfloor n/2\rfloor} + n + 5.$$ As a consequence, ... more >>>

TR15-029 | 18th February 2015
Stanislav Zak

Inherent logic and complexity

Abstract. The old intuitive question "what does the machine think" at
different stages of its computation is examined. Our paper is based on
the formal de nitions and results which are collected in the branching
program theory around the intuitive question "what does the program
know about the contents of ... more >>>

TR15-100 | 16th June 2015
Bireswar Das, Patrick Scharpfenecker, Jacobo Toran

Succinct Encodings of Graph Isomorphism

It is well known that problems encoded with circuits or formulas generally gain an exponential complexity blow-up compared to their original complexity.

We introduce a new way for encoding graph problems, based on $\textrm{CNF}$ or $\textrm{DNF}$ formulas. We show that contrary to the other existing succinct models, there are ... more >>>

TR16-007 | 23rd January 2016
Guy Kindler

Property Testing, PCP, andJuntas

Revisions: 1

The first part of this thesis strengthens the low-error PCP
characterization of NP, coming closer to the upper limit of the
conjecture of~\cite{BGLR}.

In the second part we show that a boolean function over
$n$ variables can be tested for the property of depending ... more >>>

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