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Reports tagged with Approximation Algorithms:
TR95-003 | 1st January 1995
Marek Karpinski, Alexander Zelikovsky

1.757 and 1.267-Approximation Algorithms for the Network and and Rectilinear Steiner Tree Problems

The Steiner tree problem requires to find a shortest tree connection
a given set of terminal points in a metric space. We suggest a better
and fast heuristic for the Steiner problem in graphs and in
rectilinear plane. This heuristic finds a Steiner tree at ... more >>>

TR95-023 | 16th May 1995
Sanjeev Khanna, Rajeev Motwani, Madhu Sudan, Umesh Vazirani

On Syntactic versus Computational views of Approximability

We attempt to reconcile the two distinct views of approximation
classes: syntactic and computational.
Syntactic classes such as MAX SNP allow for clean complexity-theoretic
results and natural complete problems, while computational classes such
as APX allow us to work with problems whose approximability is
well-understood. Our results give a computational ... more >>>

TR95-030 | 20th June 1995
Marek Karpinski, Alexander Zelikovsky

New Approximation Algorithms for the Steiner Tree Problems

The Steiner tree problem asks for the shortest tree connecting
a given set of terminal points in a metric space. We design
new approximation algorithms for the Steiner tree problems
using a novel technique of choosing Steiner points in dependence
on the possible deviation from ... more >>>

TR95-053 | 15th October 1995
Petr Slavik

Improved Performance of the Greedy Algorithm for the Minimum Set Cover and Minimum Partial Cover Problems

We establish significantly improved bounds on the performance of the greedy
algorithm for approximating MINIMUM SET COVER and MINIMUM PARTIAL COVER. Our
improvements result from a new approach to both problems. In particular,
(a) we improve the known bound on the performance ratio of the greedy ... more >>>

TR96-057 | 18th November 1996
Oded Goldreich, Dana Ron

Property Testing and its connection to Learning and Approximation

In this paper, we consider the question of determining whether
a function $f$ has property $P$ or is $\e$-far from any
function with property $P$.
The property testing algorithm is given a sample of the value
of $f$ on instances drawn according to some distribution.
In some cases,
more >>>

TR96-062 | 3rd December 1996
Sanjeev Khanna, Madhu Sudan, David P. Williamson

A Complete Characterization of the Approximability of Maximization Problems Derived from Boolean Constraint Satisfaction

In this paper we study the approximability of boolean constraint
satisfaction problems. A problem in this class consists of some
collection of ``constraints'' (i.e., functions
$f:\{0,1\}^k \rightarrow \{0,1\}$); an instance of a problem is a set
of constraints applied to specified subsets of $n$ boolean
variables. Schaefer earlier ... more >>>

TR96-064 | 11th December 1996
Sanjeev Khanna, Madhu Sudan, Luca Trevisan

Constraint satisfaction: The approximability of minimization problems.

This paper continues the work initiated by Creignou [Cre95] and
Khanna, Sudan and Williamson [KSW96] who classify maximization
problems derived from boolean constraint satisfaction. Here we
study the approximability of {\em minimization} problems derived
thence. A problem in this framework is characterized by a
collection F ... more >>>

TR97-001 | 8th January 1997
Marco Cesati, Luca Trevisan

On the Efficiency of Polynomial Time Approximation Schemes

A polynomial time approximation scheme (PTAS) for an optimization
problem $A$ is an algorithm that on input an instance of $A$ and
$\epsilon > 0$ finds a $(1+\epsilon)$-approximate solution in time
that is polynomial for each fixed $\epsilon$. Typical running times
are $n^{O(1/\epsilon)}$ or $2^{1/\epsilon^{O(1)}} ... 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 >>>

TR97-017 | 5th May 1997
Marek Karpinski, Juergen Wirtgen, Alexander Zelikovsky

An Approximation Algorithm for the Bandwidth Problem on Dense Graphs

The bandwidth problem is the problem of numbering the vertices of a
given graph $G$ such that the maximum difference between the numbers
of adjacent vertices is minimal. The problem has a long history and
is known to be NP-complete Papadimitriou [Pa76]. Only few special
cases ... more >>>

TR97-024 | 9th June 1997
Marek Karpinski

Polynomial Time Approximation Schemes for Some Dense Instances of NP-Hard Optimization Problems

We survey recent results on the existence of polynomial time
approximation schemes for some dense instances of NP-hard
optimization problems. We indicate further some inherent limits
for existence of such schemes for some other dense instances of
the optimization problems.

more >>>

TR97-040 | 17th September 1997
Dorit Dor, Shay Halperin, Uri Zwick

All Pairs Almost Shortest Paths

Let G=(V,E) be an unweighted undirected graph on n vertices. A simple
argument shows that computing all distances in G with an additive
one-sided error of at most 1 is as hard as Boolean matrix
multiplication. Building on recent work of Aingworth, Chekuri and
Motwani, we describe an \tilde{O}(min{n^{3/2}m^{1/2},n^{7/3}) time
more >>>

TR97-041 | 18th September 1997
Marek Karpinski, Juergen Wirtgen

On Approximation Hardness of the Bandwidth Problem

The bandwidth problem is the problem of enumerating
the vertices of a given graph $G$ such that the maximum
difference between the numbers of
adjacent vertices is minimal. The problem has a long
history and a number of applications
and is ... more >>>

TR97-056 | 1st December 1997
Oded Goldreich

Combinatorial Property Testing (a survey).

Comments: 1

We consider the question of determining whether
a given object has a predetermined property or is ``far'' from any
object having the property.
Specifically, objects are modeled by functions,
and distance between functions is measured as the fraction
of the domain on which the functions differ.
We ... more >>>

TR98-014 | 6th February 1998
Gunter Blache, Marek Karpinski, Juergen Wirtgen

On Approximation Intractability of the Bandwidth Problem

The bandwidth problem is the problem of enumerating
the vertices of a given graph $G$ such that the maximum difference
between the numbers of adjacent vertices is minimal. The problem
has a long history and a number of applications.
There was not ... more >>>

TR98-029 | 27th May 1998
Piotr Berman, Marek Karpinski

On Some Tighter Inapproximability Results

We prove a number of improved inaproximability results,
including the best up to date explicit approximation
thresholds for MIS problem of bounded degree, bounded
occurrences MAX-2SAT, and bounded degree Node Cover. We
prove also for the first time inapproximability of the
problem of Sorting by ... more >>>

TR98-031 | 4th May 1998
Dimitris Fotakis, Paul Spirakis

Graph Properties that Facilitate Travelling

In this work, we study two special cases of the metric Travelling Salesman
Problem, Graph TSP and TSP(1,2). At first, we show that dense instances of
TSP(1,2) and Graph TSP are essentially as hard to approximate as general
instances of TSP(1,2).

Next, we present an NC algorithm for TSP(1,2) that ... more >>>

TR98-065 | 6th November 1998
Piotr Berman, Marek Karpinski

On Some Tighter Inapproximability Results, Further Improvements

Improved inaproximability results are given, including the
best up to date explicit approximation thresholds for bounded
occurence satisfiability problems, like MAX-2SAT and E2-LIN-2,
and problems in bounded degree graphs, like MIS, Node Cover
and MAX CUT. We prove also for the first time inapproximability
more >>>

TR99-029 | 31st August 1999
Ilya Dumer, Daniele Micciancio, Madhu Sudan

Hardness of approximating the minimum distance of a linear code

We show that the minimum distance of a linear code (or
equivalently, the weight of the lightest codeword) is
not approximable to within any constant factor in random polynomial
time (RP), unless NP equals RP.
Under the stronger assumption that NP is not contained in RQP
(random ... more >>>

TR99-039 | 24th September 1999
Johan Hastad

On approximating CSP-B

We prove that any constraint satisfaction problem
where each variable appears a bounded number of
times admits a nontrivial polynomial time approximation

more >>>

TR00-007 | 14th December 1999
Pavlos S. Efraimidis, Paul Spirakis

Randomized Approximation Schemes for Scheduling Unrelated Parallel Machines

The problem of Scheduling $n$ Independent Jobs
on $m$ Unrelated Parallel Machines, when $m$
is fixed, is considered. The standard problem
of minimizing the makespan of the schedule
(SUM) and the bicriteria problem of scheduling
with bounded makespan and cost (SUMC), are
addressed, and randomized fully linear time
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 >>>

TR00-021 | 19th April 2000
Uriel Feige, Marek Karpinski, Michael Langberg

Improved Approximation of MAX-CUT on Graphs of Bounded Degree

We analyze the addition of a simple local improvement step to various known
randomized approximation algorithms.
Let $\alpha \simeq 0.87856$ denote the best approximation ratio currently
known for the Max Cut problem on general graphs~\cite{GW95}.
We consider a semidefinite relaxation of the Max Cut problem,
round it using the ... more >>>

TR00-043 | 21st June 2000
Uriel Feige, Marek Karpinski, Michael Langberg

A Note on Approximating MAX-BISECTION on Regular Graphs

We design a $0.795$ approximation algorithm for the Max-Bisection problem
restricted to regular graphs. In the case of three regular graphs our
results imply an approximation ratio of $0.834$.

more >>>

TR00-047 | 29th June 2000
Tobias Polzin, Siavash Vahdati Daneshmand

Primal-Dual Approaches to the Steiner Problem

We study several old and new algorithms for computing lower
and upper bounds for the Steiner problem in networks using dual-ascent
and primal-dual strategies. These strategies have been proven to be
very useful for the algorithmic treatment of the Steiner problem. We
show that none of the known algorithms ... more >>>

TR00-051 | 14th July 2000
Marek Karpinski, Miroslaw Kowaluk, Andrzej Lingas

Approximation Algorithms for MAX-BISECTION on Low Degree Regular Graphs and Planar Graphs

The max-bisection problem is to find a partition of the vertices of a
graph into two equal size subsets that maximizes the number of edges with
endpoints in both subsets.
We obtain new improved approximation ratios for the max-bisection problem on
the low degree $k$-regular graphs for ... more >>>

TR00-064 | 29th August 2000
Klaus Jansen, Marek Karpinski, Andrzej Lingas

A Polynomial Time Approximation Scheme for MAX-BISECTION on Planar Graphs

The Max-Bisection and Min-Bisection are the problems of finding
partitions of the vertices of a given graph into two equal size subsets so as
to maximize or minimize, respectively, the number of edges with exactly one
endpoint in each subset.
In this paper we design the first ... more >>>

TR00-073 | 28th August 2000
Venkatesan Guruswami, Sanjeev Khanna

On the Hardness of 4-coloring a 3-colorable Graph

We give a new proof showing that it is NP-hard to color a 3-colorable
graph using just four colors. This result is already known (Khanna,
Linial, Safra 1992), but our proof is novel as it does not rely on
the PCP theorem, while the earlier one does. This ... more >>>

TR00-079 | 12th September 2000
Mark Jerrum, Eric Vigoda

A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries

We present a fully-polynomial randomized approximation scheme
for computing the permanent of an arbitrary matrix
with non-negative entries.

more >>>

TR00-091 | 21st December 2000
Cristina Bazgan, Wenceslas Fernandez de la Vega, Marek Karpinski

Approximability of Dense Instances of NEAREST CODEWORD Problem

We give a polynomial time approximation scheme (PTAS) for dense
instances of the NEAREST CODEWORD problem.

more >>>

TR01-025 | 28th March 2001
Piotr Berman, Marek Karpinski

Approximating Minimum Unsatisfiability of Linear Equations

We consider the following optimization problem:
given a system of m linear equations in n variables over a certain field,
a feasible solution is any assignment of values to the variables, and the
minimized objective function is the number of equations that are not
satisfied. For ... more >>>

TR01-026 | 3rd April 2001
Piotr Berman, Marek Karpinski

Approximation Hardness of Bounded Degree MIN-CSP and MIN-BISECTION

We consider bounded occurrence (degree) instances of a minimum
constraint satisfaction problem MIN-LIN2 and a MIN-BISECTION problem for
graphs. MIN-LIN2 is an optimization problem for a given system of linear
equations mod 2 to construct a solution that satisfies the minimum number
of them. E3-OCC-MIN-E3-LIN2 ... more >>>

TR01-034 | 30th April 2001
Cristina Bazgan, Wenceslas Fernandez de la Vega, Marek Karpinski

Polynomial Time Approximation Schemes for Dense Instances of Minimum Constraint Satisfaction

It is known that large fragments of the class of dense
Minimum Constraint Satisfaction (MIN-CSP) problems do not have
polynomial time approximation schemes (PTASs) contrary to their
Maximum Constraint Satisfaction analogs. In this paper we prove,
somewhat surprisingly, that the minimum satisfaction of dense
instances of kSAT-formulas, ... more >>>

TR01-042 | 31st May 2001
Marek Karpinski

Approximating Bounded Degree Instances of NP-Hard Problems

We present some of the recent results on computational complexity
of approximating bounded degree combinatorial optimization problems. In
particular, we present the best up to now known explicit nonapproximability
bounds on the very small degree optimization problems which are of
particular importance on the intermediate stages ... more >>>

TR01-047 | 3rd July 2001
Piotr Berman, Sridhar Hannenhalli, Marek Karpinski

1.375-Approximation Algorithm for Sorting by Reversals

Analysis of genomes evolving by inversions leads to a general
combinatorial problem of {\em Sorting by Reversals}, MIN-SBR, the problem of
sorting a permutation by a minimum number of reversals.
This combinatorial problem has a long history, and a number of other
motivations. It was studied in a great ... more >>>

TR01-053 | 17th July 2001
Piotr Berman, Marek Karpinski

Efficient Amplifiers and Bounded Degree Optimization

This paper studies the existence of efficient (small size)
amplifiers for proving explicit inaproximability results for bounded degree
and bounded occurrence combinatorial optimization problems, and gives
an explicit construction for such amplifiers. We use this construction
also later to improve the currently best known approximation lower bounds
more >>>

TR01-063 | 5th August 2001
Michal Parnas, Dana Ron, Alex Samorodnitsky

Proclaiming Dictators and Juntas or Testing Boolean Formulae

Revisions: 1

We consider the problem of determining whether a given
function f : {0,1}^n -> {0,1} belongs to a certain class
of Boolean functions F or whether it is far from the class.
More precisely, given query access to the function f and given
a distance parameter epsilon, we would ... more >>>

TR01-097 | 11th December 2001
Piotr Berman, Marek Karpinski

Improved Approximations for General Minimum Cost Scheduling

We give improved trade-off results on approximating general
minimum cost scheduling problems.

more >>>

TR01-100 | 14th December 2001
Noga Alon, Wenceslas Fernandez de la Vega, Ravi Kannan, Marek Karpinski

Random Sampling and Approximation of MAX-CSP Problems

We present a new efficient sampling method for approximating
r-dimensional Maximum Constraint Satisfaction Problems, MAX-rCSP, on
n variables up to an additive error \epsilon n^r.We prove a new
general paradigm in that it suffices, for a given set of constraints,
to pick a small uniformly random ... more >>>

TR02-018 | 22nd March 2002
Piotr Berman, Marek Karpinski, Yakov Nekrich

Approximating Huffman Codes in Parallel

In this paper we present some new results on the approximate parallel
construction of Huffman codes. Our algorithm achieves linear work
and logarithmic time, provided that the initial set of elements
is sorted. This is the first parallel algorithm for that problem
with the optimal time and ... more >>>

TR02-041 | 2nd July 2002
Wenceslas Fernandez de la Vega, Marek Karpinski, Claire Kenyon

A Polynomial Time Approximation Scheme for Metric MIN-BISECTION

We design a polynomial time approximation scheme (PTAS) for
the problem of Metric MIN-BISECTION of dividing a given finite metric
space into two halves so as to minimize the sum of distances across
that partition. The method of solution depends on a new metric placement
partitioning ... more >>>

TR02-044 | 16th July 2002
Wenceslas Fernandez de la Vega, Marek Karpinski

A Polynomial Time Approximation Scheme for Subdense MAX-CUT

We prove that the subdense instances of MAX-CUT of average
degree Omega(n/logn) posses a polynomial time approximation scheme (PTAS).
We extend this result also to show that the instances of general 2-ary
maximum constraint satisfaction problems (MAX-CSP) of the same average
density have PTASs. Our results ... more >>>

TR02-046 | 16th July 2002
Marek Karpinski

On Approximability of Minimum Bisection Problem

We survey some recent results on the complexity of computing
approximate solutions for instances of the Minimum Bisection problem
and formulate some intriguing and still open questions about the
approximability status of that problem. Some connections to other
optimization problems are also indicated.

more >>>

TR02-070 | 13th December 2002
Wenceslas Fernandez de la Vega, Marek Karpinski

9/8-Approximation Algorithm for Random MAX-3SAT

Revisions: 1

We prove that MAX-3SAT can be approximated in polynomial time
within a factor 9/8 on random instances.

more >>>

TR02-073 | 12th December 2002
Janka Chlebíková, Miroslav Chlebík

Approximation Hardness for Small Occurrence Instances of NP-Hard Problem

The paper contributes to the systematic study (started by Berman and
Karpinski) of explicit approximability lower bounds for small occurrence optimization
problems. We present parametrized reductions for some packing and
covering problems, including 3-Dimensional Matching, and prove the best
known inapproximability results even for highly restricted versions of ... more >>>

TR03-008 | 11th February 2003
Piotr Berman, Marek Karpinski

Improved Approximation Lower Bounds on Small Occurrence Optimization

We improve a number of approximation lower bounds for
bounded occurrence optimization problems like MAX-2SAT,
E2-LIN-2, Maximum Independent Set and Maximum-3D-Matching.

more >>>

TR03-031 | 8th April 2003
Birgit Schelm

Average-Case Complexity Theory of Approximation Problems

Both average-case complexity and the study of the approximability properties of NP-optimization problems are well established and active fields of research. By applying the notion of average-case complexity to approximation problems we provide a formal framework that allows the classification of NP-optimization problems according to their average-case approximability. Thus, known ... more >>>

TR03-032 | 16th April 2003
Andreas Björklund, Thore Husfeldt, Sanjeev Khanna

Approximating Longest Directed Path

We investigate the hardness of approximating the longest path and
the longest cycle in directed graphs on $n$ vertices. We show that
neither of these two problems can be polynomial time approximated
within $n^{1-\epsilon}$ for any $\epsilon>0$ unless
$\text{P}=\text{NP}$. In particular, the result holds for
more >>>

TR03-035 | 21st May 2003
Eran Halperin, Guy Kortsarz, Robert Krauthgamer

Tight lower bounds for the asymmetric k-center problem

In the {\sc $k$-center} problem, the input is a bound $k$
and $n$ points with the distance between every two of them,
such that the distances obey the triangle inequality.
The goal is to choose a set of $k$ points to serve as centers,
so that the maximum distance ... more >>>

TR03-038 | 15th May 2003
Julia Chuzhoy, Sudipto Guha, Sanjeev Khanna, Seffi Naor

Asymmetric k-center is log^*n-hard to Approximate

We show that the asymmetric $k$-center problem is
$\Omega(\log^* n)$-hard to approximate unless
${\rm NP} \subseteq {\rm DTIME}(n^{poly(\log \log n)})$.
Since an $O(\log^* n)$-approximation algorithm is known
for this problem, this essentially resolves the approximability
of this problem. This is the first natural problem
whose approximability threshold does not polynomially ... more >>>

TR03-056 | 29th July 2003
Piotr Berman, Marek Karpinski

Approximability of Hypergraph Minimum Bisection

We prove that the problems of minimum bisection on k-uniform
hypergraphs are almost exactly as hard to approximate,
up to the factor k/3, as the problem of minimum bisection
on graphs. On a positive side, our argument gives also the
first approximation ... more >>>

TR04-010 | 26th January 2004
Michal Parnas, Dana Ron, Ronitt Rubinfeld

Tolerant Property Testing and Distance Approximation

A standard property testing algorithm is required to determine
with high probability whether a given object has property
P or whether it is \epsilon-far from having P, for any given
distance parameter \epsilon. An object is said to be \epsilon-far
from having ... more >>>

TR04-051 | 10th June 2004
Zdenek Dvorák, Daniel Král, Ondrej Pangrác

Locally consistent constraint satisfaction problems

An instance of a constraint satisfaction problem is $l$-consistent
if any $l$ constraints of it can be simultaneously satisfied.
For a set $\Pi$ of constraint types, $\rho_l(\Pi)$ denotes the largest ratio of constraints which can be satisfied in any $l$-consistent instance composed by constraints from the set $\Pi$. In the ... more >>>

TR04-119 | 8th December 2004
Uriel Feige, Daniel Reichman

On The Hardness of Approximating Max-Satisfy

Max-Satisfy is the problem of finding an assignment that satisfies
the maximum number of equations in a system of linear equations
over $\mathbb{Q}$. We prove that unless NP$\subseteq $BPP there is no
polynomial time algorithm for the problem achieving an
approximation ratio of $1/n^{1-\epsilon}$, where $n$ is the number
of ... more >>>

TR05-029 | 2nd March 2005
Frank Neumann, Marco Laumanns

Speeding Up Approximation Algorithms for NP-hard Spanning Forest Problems by Multi-objective Optimization

Revisions: 1

We give faster approximation algorithms for the
generalization of two NP-hard spanning tree problems. First,
we investigate the problem of minimizing the degree of
minimum spanning forests. The task is to compute for each
number of connected components a minimum spanning forest
whose degree is as small as possible. Fischer
more >>>

TR05-069 | 11th July 2005
Piotr Berman, Marek Karpinski

8/7-Approximation Algorithm for (1,2)-TSP

Revisions: 2

We design a polynomial time 8/7-approximation algorithm for the Traveling Salesman Problem in which all distances are either one or two. This improves over the best known approximation factor of 7/6 for that problem. As a direct application we get a 7/6-approximation algorithm for the Maximum Path Cover Problem, similarily ... more >>>

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 >>>

TR05-125 | 2nd November 2005
Sofya Raskhodnikova, Dana Ron, Ronitt Rubinfeld, Amir Shpilka, Adam Smith

Sublinear Algorithms for Approximating String Compressibility and the Distribution Support Size

We raise the question of approximating compressibility of a string with respect to a fixed compression scheme, in sublinear time. We study this question in detail for two popular lossless compression schemes: run-length encoding (RLE) and Lempel-Ziv (LZ), and present algorithms and lower bounds for approximating compressibility with respect to ... more >>>

TR05-131 | 7th November 2005
Don Coppersmith, Lisa Fleischer, Atri Rudra

Ordering by weighted number of wins gives a good ranking for weighted tournaments

We consider the following simple algorithm for feedback arc set problem in weighted tournaments --- order the vertices by their weighted indegrees. We show that this algorithm has an approximation guarantee of $5$ if the weights satisfy \textit{probability constraints}
(for any pair of vertices $u$ and $v$, $w_{uv}+w_{vu}=1$). Special cases ... more >>>

TR06-045 | 13th March 2006
Jan Arpe, Bodo Manthey

Approximability of Minimum AND-Circuits

Revisions: 1

Given a set of monomials, the Minimum AND-Circuit problem asks for a
circuit that computes these monomials using AND-gates of fan-in two and
being of minimum size. We prove that the problem is not polynomial time
approximable within a factor of less than 1.0051 unless P = NP, even if
more >>>

TR06-101 | 22nd August 2006
Wenceslas Fernandez de la Vega, Marek Karpinski

Approximation Complexity of Nondense Instances of MAX-CUT

We prove existence of approximation schemes for instances of MAX-CUT with $\Omega(\frac{n^2}{\Delta})$ edges which work in $2^{O^\thicksim(\frac{\Delta}{\varepsilon^2})}n^{O(1)}$ time. This entails in particular existence of quasi-polynomial approximation schemes (QPTASs) for mildly sparse instances of MAX-CUT with $\Omega(\frac{n^2}{\operatorname{polylog} n})$ edges. The result depends on new sampling method for smoothed linear programs that ... more >>>

TR06-124 | 25th September 2006
Wenceslas Fernandez de la Vega, Ravi Kannan, Marek Karpinski

Approximation of Global MAX-CSP Problems

We study the problem of absolute approximability of MAX-CSP problems with the global constraints. We prove existence of an efficient sampling method for the MAX-CSP class of problems with linear global constraints and bounded feasibility gap. It gives for the first time a polynomial in epsilon^-1 sample complexity bound for ... 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 >>>

TR07-011 | 19th December 2006
Bodo Manthey

On Approximating Restricted Cycle Covers

A cycle cover of a graph is a set of cycles such that every vertex is
part of exactly one cycle. An L-cycle cover is a cycle cover in which
the length of every cycle is in the set L. The weight of a cycle cover
of an edge-weighted graph ... more >>>

TR07-101 | 10th September 2007
Patrick Briest, Martin Hoefer, Piotr Krysta

Stackelberg Network Pricing Games

Revisions: 1

We study a multi-player one-round game termed Stackelberg Network Pricing Game, in which a leader can set prices for a subset of m pricable edges in a graph. The other edges have a fixed cost. Based on the leader's decision one or more followers optimize a polynomial-time solvable combinatorial minimization ... more >>>

TR07-106 | 10th September 2007
Yijia Chen, Martin Grohe, Magdalena Grüber

On Parameterized Approximability

Combining classical approximability questions with parameterized complexity, we introduce a theory of parameterized approximability.
The main intention of this theory is to deal with the efficient approximation of small cost solutions for optimisation problems.

more >>>

TR07-119 | 5th December 2007
Piotr Berman, Bhaskar DasGupta, Marek Karpinski

Approximating Transitive Reductions for Directed Networks

We consider <i>minimum equivalent digraph</i> (<i>directed network</i>) problem (also known as the <i>strong transitive reduction</i>), its maximum optimization variant, and some extensions of those two types of problems. We prove the existence of polynomial time approximation algorithms with ratios 1.5 for all the minimization problems and 2 for all the ... more >>>

TR08-094 | 10th October 2008
Piotr Berman, Marek Karpinski, Alexander Zelikovsky

1.25 Approximation Algorithm for the Steiner Tree Problem with Distances One and Two

We give a 1.25 approximation algorithm for the Steiner Tree Problem with distances one and two, improving on the best known bound for that problem.

more >>>

TR08-101 | 20th November 2008
Marek Karpinski, Warren Schudy

Linear Time Approximation Schemes for the Gale-Berlekamp Game and Related Minimization Problems

We design a linear time approximation scheme for the Gale-Berlekamp Switching Game and generalize it to a wider class of dense fragile minimization problems including the Nearest Codeword Problem (NCP) and Unique Games Problem. Further applications include, among other things, finding a constrained form of matrix rigidity and maximum likelihood ... more >>>

TR09-076 | 19th August 2009
Christian Glaßer, Christian Reitwießner, Maximilian Witek

Improved and Derandomized Approximations for Two-Criteria Metric Traveling Salesman

Revisions: 1

We improve and derandomize the best known approximation algorithm for the two-criteria metric traveling salesman problem (2-TSP). More precisely, we construct a deterministic 2-approximation which answers an open question by Manthey.

Moreover, we show that 2-TSP is randomized $(3/2+\epsilon ,2)$-approximable, and we give the first randomized approximations for the two-criteria ... more >>>

TR10-041 | 11th March 2010
Sanjeev Arora, Russell Impagliazzo, William Matthews, David Steurer

Improved Algorithms for Unique Games via Divide and Conquer

We present two new approximation algorithms for Unique Games. The first generalizes the results of Arora, Khot, Kolla, Steurer, Tulsiani, and Vishnoi who give polynomial time approximation algorithms for graphs with high conductance. We give a polynomial time algorithm assuming only good local conductance, i.e. high conductance for small subgraphs. ... more >>>

TR10-132 | 18th August 2010
Mahdi Cheraghchi, Johan Hastad, Marcus Isaksson, Ola Svensson

Approximating Linear Threshold Predicates

We study constraint satisfaction problems on the domain $\{-1,1\}$, where the given constraints are homogeneous linear threshold predicates. That is, predicates of the form $\mathrm{sgn}(w_1 x_1 + \cdots + w_n x_n)$ for some positive integer weights $w_1, \dots, w_n$. Despite their simplicity, current techniques fall short of providing a classification ... more >>>

TR11-156 | 23rd November 2011
Marek Karpinski, Richard Schmied

Improved Lower Bounds for the Shortest Superstring and Related Problems

Revisions: 1

We study the approximation hardness of the Shortest Superstring, the Maximal Compression and
the Maximum Asymmetric Traveling Salesperson (MAX-ATSP) problem.
We introduce a new reduction method that produces strongly restricted instances of
the Shortest Superstring problem, in which the maximal orbit size is eight
(with no ... more >>>

TR12-008 | 30th January 2012
Marek Karpinski, Richard Schmied

On Approximation Lower Bounds for TSP with Bounded Metrics

Revisions: 1

We develop a new method for proving explicit approximation lower bounds for TSP problems with bounded metrics improving on the best up to now known bounds. They almost match the best known bounds for unbounded metric TSP problems. In particular, we prove the best known lower bound for TSP with ... more >>>

TR12-170 | 30th November 2012
Scott Aaronson, Travis Hance

Generalizing and Derandomizing Gurvits's Approximation Algorithm for the Permanent

Around 2002, Leonid Gurvits gave a striking randomized algorithm to approximate the permanent of an n×n matrix A. The algorithm runs in O(n^2/?^2) time, and approximates Per(A) to within ±?||A||^n additive error. A major advantage of Gurvits's algorithm is that it works for arbitrary matrices, not just for nonnegative matrices. ... more >>>

TR13-045 | 26th March 2013
Marek Karpinski, Michael Lampis, Richard Schmied

New Inapproximability Bounds for TSP

In this paper, we study the approximability of the metric Traveling Salesman Problem, one of the most widely studied problems in combinatorial optimization. Currently, the best known hardness of approximation bounds are 185/184 for the symmetric case (due to Lampis) and 117/116 for the asymmetric case (due to Papadimitriou and ... more >>>

TR13-066 | 25th April 2013
Marek Karpinski, Richard Schmied

Approximation Hardness of Graphic TSP on Cubic Graphs

We prove explicit approximation hardness results for the Graphic TSP on cubic and subcubic graphs as well as the new inapproximability bounds for the corresponding instances of the (1,2)-TSP. The proof technique uses new modular constructions of simulating gadgets for the restricted cubic and subcubic instances. The modular constructions used ... 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 >>>

TR15-097 | 16th June 2015
Marek Karpinski

Towards Better Inapproximability Bounds for TSP: A Challenge of Global Dependencies

We present in this paper some of the recent techniques and methods for proving best up to now explicit approximation hardness bounds for metric symmetric and asymmetric Traveling Salesman Problem (TSP) as well as related problems of Shortest Superstring and Maximum Compression. We attempt to shed some light on the ... more >>>

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