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Reports tagged with Vertex Cover:
TR95-024 | 23rd May 1995
Mihir Bellare, Oded Goldreich, Madhu Sudan

Free bits, PCP and Non-Approximability - Towards tight results

Revisions: 4

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 >>>

TR97-004 | 19th February 1997
Marek Karpinski, Alexander Zelikovsky

Approximating Dense Cases of Covering Problems

Comments: 1

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 ... more >>>

TR01-094 | 3rd December 2001
Jonas Holmerin

Vertex Cover on 4-regular Hyper-graphs is Hard to Approximate Within 2 - \epsilon

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 ... more >>>

TR01-102 | 18th December 2001
Oded Goldreich

Using the FGLSS-reduction to Prove Inapproximability Results for Minimum Vertex Cover in Hypergraphs.

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 ... more >>>

TR01-104 | 17th December 2001
Irit Dinur, Shmuel Safra

The Importance of Being Biased

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.

more >>>

TR02-027 | 30th April 2002
Irit Dinur, Venkatesan Guruswami, Subhash Khot

Vertex Cover on k-Uniform Hypergraphs is Hard to Approximate within Factor (k-3-\epsilon)

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 >>>

TR04-084 | 28th September 2004
George Karakostas

A better approximation ratio for the Vertex Cover problem

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 ... more >>>

TR04-101 | 28th September 2004
Miroslav Chlebik, Janka Chlebíková

Crown reductions for the Minimum Weighted Vertex Cover problem

TR05-094 | 9th August 2005
Michal Parnas, Dana Ron

On Approximating the Minimum Vertex Cover in Sublinear Time and the Connection to Distributed Algorithms

Revisions: 1

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 ... more >>>

TR05-141 | 29th November 2005
Amos Beimel, Paz Carmi, Kobbi Nissim, Enav Weinreb

Private Approximation of Search Problems

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 ... more >>>

TR06-098 | 17th August 2006
Grant Schoenebeck, Luca Trevisan, Madhur Tulsiani

A Linear Round Lower Bound for Lovasz-Schrijver SDP Relaxations of Vertex Cover

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 ... more >>>

TR06-152 | 6th December 2006
Konstantinos Georgiou, Avner Magen, Iannis Tourlakis

Tight integrality gaps for Vertex Cover SDPs in the Lovasz-Schrijver hierarchy

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 >>>

TR10-038 | 10th March 2010
Dieter van Melkebeek, Holger Dell

Satisfiability Allows No Nontrivial Sparsification Unless The Polynomial-Time Hierarchy Collapses

Revisions: 1

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 >>>

TR10-169 | 10th November 2010
Siavosh Benabbas, Konstantinos Georgiou, Avner Magen

The Sherali-Adams System Applied to Vertex Cover: Why Borsuk Graphs Fool Strong LPs and some Tight Integrality Gaps for SDPs

Revisions: 2

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 ... more >>>

TR11-019 | 5th February 2011
Valentin Brimkov, Andrew Leach, Jimmy Wu, Michael Mastroianni

On the Approximability of a Geometric Set Cover Problem

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 >>>

TR11-098 | 11th July 2011
Marek Karpinski, Richard Schmied, Claus Viehmann

Tight Approximation Bounds for Vertex Cover on Dense k-Partite Hypergraphs

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

more >>>

TR15-157 | 1st September 2015
Thomas O'Neil

Representation-Independent Fixed Parameter Tractability for Vertex Cover and Weighted Monotone Satisfiability

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 >>>

TR16-124 | 12th August 2016
Subhash Khot

On Independent Sets, $2$-to-$2$ Games and Grassmann Graphs

Revisions: 1 , Comments: 1

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 ... more >>>

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