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 >>>

Marek Karpinski, Alexander Zelikovsky

We study dense instances of several covering problems. An instance of

the set cover problem with $m$ sets is dense if there is $\epsilon>0$

such that any element belongs to at least $\epsilon m$ sets. We show

that the dense set cover problem can be approximated with ...
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Jonas Holmerin

We prove that Minimum vertex cover on 4-regular hyper-graphs (or

in other words, hitting set where all sets have size exactly 4),

is hard to approximate within 2 - \epsilon.

We also prove that the maximization version, in which we

are allowed to pick ...
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Oded Goldreich

Using known results regarding PCP,

we present simple proofs of the inapproximability

of vertex cover for hypergraphs.

Specifically, we show that

(1) Approximating the size of the minimum vertex cover

in $O(1)$-regular hypergraphs to within a factor of~1.99999 is NP-hard.

(2) Approximating the size ...
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Irit Dinur, Shmuel Safra

We show Minimum Vertex Cover NP-hard to approximate to within a factor

of 1.3606. This improves on the previously known factor of 7/6.

Irit Dinur, Venkatesan Guruswami, Subhash Khot

Given a $k$-uniform hypergraph, the E$k$-Vertex-Cover problem is

to find a minimum subset of vertices that ``hits'' every edge. We

show that for every integer $k \geq 5$, E$k$-Vertex-Cover is

NP-hard to approximate within a factor of $(k-3-\epsilon)$, for

an arbitrarily small constant $\epsilon > 0$.

This almost matches the ... more >>>

George Karakostas

We reduce the approximation factor for Vertex Cover to $2-\Theta(1/\sqrt{logn})$

(instead of the previous $2-\Theta(loglogn/logn})$, obtained by Bar-Yehuda and Even,

and by Monien and Speckenmeyer in 1985. The improvement of the vanishing

factor comes as an application of the recent results of Arora, Rao, and Vazirani

that improved ...
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Miroslav Chlebik, Janka Chlebíková

Michal Parnas, Dana Ron

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 ...
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Amos Beimel, Paz Carmi, Kobbi Nissim, Enav Weinreb

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 ...
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Grant Schoenebeck, Luca Trevisan, Madhur Tulsiani

We study semidefinite programming relaxations of Vertex Cover arising from

repeated applications of the LS+ ``lift-and-project'' method of Lovasz and

Schrijver starting from the standard linear programming relaxation.

Goemans and Kleinberg prove that after one round of LS+ the integrality

gap remains arbitrarily close to 2. Charikar proves an integrality ...
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Konstantinos Georgiou, Avner Magen, Iannis Tourlakis

We prove that the integrality gap after tightening the standard LP relaxation for Vertex Cover with Omega(sqrt(log n/log log n)) rounds of the SDP LS+ system is 2-o(1).

more >>>Dieter van Melkebeek, Holger Dell

Consider the following two-player communication process to decide a language $L$: The first player holds the entire input $x$ but is polynomially bounded; the second player is computationally unbounded but does not know any part of $x$; their goal is to cooperatively decide whether $x$ belongs to $L$ at small ... more >>>

Siavosh Benabbas, Konstantinos Georgiou, Avner Magen

We study the performance of the Sherali-Adams system for VERTEX COVER on graphs with vector

chromatic number $2+\epsilon$. We are able to construct solutions for LPs derived by any number of Sherali-Adams tightenings by introducing a new tool to establish Local-Global Discrepancy. When restricted to

$O(1/ \epsilon)$ tightenings we show ...
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Valentin Brimkov, Andrew Leach, Jimmy Wu, Michael Mastroianni

Given a finite set of straight line segments $S$ in $R^{2}$ and some $k\in N$, is there a subset $V$ of points on segments in $S$ with $\vert V \vert \leq k$ such that each segment of $S$ contains at least one point in $V$? This is a special case ... more >>>

Marek Karpinski, Richard Schmied, Claus Viehmann

We establish almost tight upper and lower approximation bounds for the Vertex Cover problem on dense k-partite hypergraphs.

more >>>Thomas O'Neil

A symmetric representation for a set of objects requires the same amount of space for the set as for its complement. Complexity classifications that are based on the length of the input can depend on whether the representation is symmetric. In this article we describe a symmetric representation scheme for ... more >>>

Subhash Khot

We present a candidate reduction from the $3$-Lin problem to the $2$-to-$2$ Games problem and present a combinatorial hypothesis about

Grassmann graphs which, if correct, is sufficient to show the soundness of the reduction in

a certain non-standard sense. A reduction that is sound in this non-standard sense

implies that ...
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Max Bannach, Zacharias Heinrich, Rüdiger Reischuk, Till Tantau

Computing kernels for the hitting set problem (the problem of

finding a size-$k$ set that intersects each hyperedge of a

hypergraph) is a well-studied computational problem. For hypergraphs

with $m$ hyperedges, each of size at most~$d$, the best algorithms

can compute kernels of size $O(k^d)$ in ...
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