Mihir Bellare, Oded Goldreich, Madhu Sudan

This paper continues the investigation of the connection between proof

systems and approximation. The emphasis is on proving ``tight''

non-approximability results via consideration of measures like the

``free bit complexity'' and the ``amortized free bit complexity'' of

proof systems.

The first part of the paper presents a collection of new ... more >>>

Günter Hotz, Gero Vierke, Bjoern Schieffer

In this paper the $R$-machines defined by Blum, Shub and Smale

are generalized by allowing infinite convergent computations.

The description of real numbers is infinite.

Therefore, considering arithmetic operations on real numbers should

also imply infinite computations on {\em analytic machines}.

We prove that $\R$-computable functions are $\Q$-analytic.

We show ...
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Jin-Yi Cai, Ajay Nerurkar

Recently Ajtai showed that

to approximate the shortest lattice vector in the $l_2$-norm within a

factor $(1+2^{-\mbox{\tiny dim}^k})$, for a sufficiently large

constant $k$, is NP-hard under randomized reductions.

We improve this result to show that

to approximate a shortest lattice vector within a

factor $(1+ \mbox{dim}^{-\epsilon})$, for any

$\epsilon>0$, ...
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Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, Mario Szegedy

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 ...
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Bruce Edward Litow

It is shown that integer coprimality is in NC.

more >>>Phong Nguyen, Jacques Stern

Recently, Ajtai discovered a fascinating connection

between the worst-case complexity and the average-case

complexity of some well-known lattice problems.

Later, Ajtai and Dwork proposed a cryptosystem inspired

by Ajtai's work, provably secure if a particular lattice

problem is difficult. We show that there is a converse

to the ...
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Venkatesan Guruswami, Daniel Lewin and Madhu Sudan, Luca Trevisan

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 ...
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Madhu Sudan, Luca Trevisan

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 ...
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Irit Dinur, Eldar Fischer, Guy Kindler, Ran Raz, Shmuel Safra

This paper strengthens the low-error PCP characterization of NP, coming

closer to the ultimate BGLR conjecture. Namely, we prove that witnesses for

membership in any NP language can be verified with a constant

number of accesses, and with an error probability exponentially

small in the ...
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Matthias Krause, Petr Savicky, Ingo Wegener

Ordered binary decision diagrams (OBDDs) and their variants

are motivated by the need to represent Boolean functions

in applications. Research concerning these applications leads

also to problems and results interesting from theoretical

point of view. In this paper, methods from communication

complexity and ...
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Lars Engebretsen

We show that the k-CSP problem over a finite Abelian group G

cannot be approximated within |G|^{k-O(sqrt{k})}-epsilon, for

any constant epsilon>0, unless P=NP. This lower bound matches

well with the best known upper bound, |G|^{k-1}, of Serna,

Trevisan and Xhafa. The proof uses a combination of PCP

techniques---most notably a ...
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Martin Sauerhoff

This paper deals with the number of monochromatic combinatorial

rectangles required to approximate a Boolean function on a constant

fraction of all inputs, where each rectangle may be defined with

respect to its own partition of the input variables. The main result

of the paper is that the number of ...
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Andreas Jakoby, Maciej Liskiewicz, Rüdiger Reischuk

A model for parallel and distributed programs, the dynamic process graph (DPG),

is investigated under graph-theoretic and complexity aspects.

Such graphs embed constructors for parallel programs,

synchronization mechanisms as well as conditional branches.

They are capable of representing all possible executions of a

parallel or distributed program ...
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Lars Engebretsen, Jonas Holmerin, Alexander Russell

An equation over a finite group G is an expression of form

w_1 w_2...w_k = 1_G, where each w_i is a variable, an inverted

variable, or a constant from G; such an equation is satisfiable

if there is a setting of the variables to values in G ...
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Lars Engebretsen, Jonas Holmerin

We study non-Boolean PCPs that have perfect completeness and read

three positions from the proof. For the case when the proof consists

of values from a domain of size d for some integer constant d

>= 2, we construct a non-adaptive PCP with perfect completeness

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Elad Hazan, Shmuel Safra, Oded Schwartz

We study bounded degree graph problems, mainly the problem of

k-Dimensional Matching \emph{(k-DM)}, namely, the problem of

finding a maximal matching in a k-partite k-uniform balanced

hyper-graph. We prove that k-DM cannot be efficiently approximated

to within a factor of $ O(\frac{k}{ \ln k}) $ unless $P = NP$.

This ...
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Till Tantau

This paper introduces logspace optimisation problems as

analogues of the well-studied polynomial-time optimisation

problems. Similarly to them, logspace

optimisation problems can have vastly different approximation

properties, even though the underlying existence and budget problems

have the same computational complexity. Numerous natural problems

are presented that exhibit such a varying ...
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Andrzej Lingas, Martin Wahlén

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 >>>

André Lanka, Andreas Goerdt

Assuming 3-SAT formulas are hard to refute

on average, Feige showed some approximation hardness

results for several problems like min bisection, dense

$k$-subgraph, max bipartite clique and the 2-catalog segmentation

problem. We show a similar result for

max bipartite clique, but under the assumption, 4-SAT formulas

are hard to refute ...
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Oded Lachish, Ilan Newman

A string $\alpha\in\Sigma^n$ is called {\it p-periodic},

if for every $i,j \in \{1,\dots,n\}$, such that $i\equiv j \bmod p$,

$\alpha_i = \alpha_{j}$, where $\alpha_i$ is the $i$-th place of $\alpha$.

A string $\alpha\in\Sigma^n$ is said to be $period(\leq g)$,

if there exists $p\in \{1,\dots,g\}$ such that $\alpha$ ...
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Sanjeev Arora, Eli Berger, Elad Hazan, Guy Kindler, Muli Safra

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 ...
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Oded Goldreich, Dana Ron

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, ...
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Oded Goldreich

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,

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Piotr Berman, Jieun Jeong, Shiva Prasad Kasiviswanathan, Bhuvan Urgaonkar

In our problem we are given a set of customers, their positions on the

plane and their demands. Geometrically, directional antenna with

parameters $\alpha,\rho,R$ is a set

of points with radial coordinates $(\theta,r)$ such that

$\alpha \le \theta \le \alpha+\rho$ and $r \le R$. Given a set of

possible directional ...
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Eldar Fischer, Orly Yahalom

A coloring of a graph is {\it convex} if it

induces a partition of the vertices into connected subgraphs.

Besides being an interesting property from a theoretical point of

view, tests for convexity have applications in various areas

involving large graphs. Our results concern the important subcase

of testing for ...
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Patrick Briest, Piotr Krysta

We investigate non-parametric unit-demand pricing problems, in which the goal is to find revenue maximizing prices for a set of products based on consumer profiles obtained, e.g., from an e-Commerce website. A consumer profile consists of a number of non-zero budgets and a ranking of all the products the consumer ... more >>>

Sharon Feldman, Guy Kortsarz, Zeev Nutov

We consider the k-Directed Steiner Forest} (k-dsf) problem:

given a directed graph G=(V,E) with edge costs, a collection D subseteq V \times V

of ordered node pairs, and an integer k leq |D|, find a minimum cost subgraph

H of G

that contains an st-path for (at least) k ...
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Richard Beigel, Bin Fu

The bin packing problem is to find the minimum

number of bins of size one to pack a list of items with sizes

$a_1,\ldots , a_n$ in $(0,1]$. Using uniform sampling, which selects

a random element from the input list each time, we develop a

randomized $O({n(\log n)(\log\log n)\over ...
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Ilan Komargodski, Ran Raz

We give an explicit function $h:\{0,1\}^n\to\{0,1\}$ such that any deMorgan formula of size $O(n^{2.499})$ agrees with $h$ on at most $\frac{1}{2} + \epsilon$ fraction of the inputs, where $\epsilon$ is exponentially small (i.e. $\epsilon = 2^{-n^{\Omega(1)}}$). Previous lower bounds for formula size were obtained for exact computation.

The same ... more >>>

Eric Blais, Li-Yang Tan

We study the complexity of approximating Boolean functions with DNFs and other depth-2 circuits, exploring two main directions: universal bounds on the approximability of all Boolean functions, and the approximability of the parity function.

In the first direction, our main positive results are the first non-trivial universal upper bounds on ...
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Paul Goldberg, Aaron Roth

We analyze the number of payoff queries needed to compute approximate correlated equilibria. For multi-player, binary-choice games, we show logarithmic upper and lower bounds on the query complexity of approximate correlated equilibrium. For well-supported approximate correlated equilibrium (a restriction where a player's behavior must always be approximately optimal, in the ... more >>>

Anna Gal, Jing-Tang Jang, Nutan Limaye, Meena Mahajan, Karteek Sreenivasaiah

SUBSET SUM is a well known NP-complete problem:

given $t \in Z^{+}$ and a set $S$ of $m$ positive integers, output YES if and only if there is a subset $S^\prime \subseteq S$ such that the sum of all numbers in $S^\prime$ equals $t$. The problem and its search ...
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Igor Carboni Oliveira, Rahul Santhanam

We continue the study of pseudo-deterministic algorithms initiated by Gat and Goldwasser

[GG11]. A pseudo-deterministic algorithm is a probabilistic algorithm which produces a fixed

output with high probability. We explore pseudo-determinism in the settings of learning and ap-

proximation. Our goal is to simulate known randomized algorithms in these settings ...
more >>>