Andrea E. F. Clementi, Luca Trevisan

We provide new non-approximability results for the restrictions

of the min-VC problem to bounded-degree, sparse and dense graphs.

We show that for a sufficiently large B, the recent 16/15 lower

bound proved by Bellare et al. extends with negligible

loss to graphs with bounded ...
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Sanjeev Khanna, Madhu Sudan

In 1978, Schaefer considered a subclass of languages in

NP and proved a ``dichotomy theorem'' for this class. The subclass

considered were problems expressible as ``constraint satisfaction

problems'', and the ``dichotomy theorem'' showed that every language in

this class is either in P, or is NP-hard. This result is in ...
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Marek Karpinski

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.

Marek Karpinski, Juergen Wirtgen

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 ...
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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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Gunter Blache, Marek Karpinski, Juergen Wirtgen

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 ...
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Uriel Feige, Marek Karpinski, Michael Langberg

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 $0.834$.

Andrea E. F. Clementi, Paolo Penna, Riccardo Silvestri

Given a finite set $S$ of points (i.e. the stations of a radio

network) on a $d$-dimensional Euclidean space and a positive integer

$1\le h \le |S|-1$, the \minrangeh{d} problem

consists of assigning transmission ranges to the stations so as

to minimize the total power consumption, provided ...
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Klaus Jansen, Marek Karpinski, Andrzej Lingas

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 ...
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Venkatesan Guruswami, Sanjeev Khanna

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 ...
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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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Wenceslas Fernandez de la Vega, Marek Karpinski

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 ...
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Marek Karpinski

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.

Piotr Berman, Marek Karpinski, Alexander D. Scott, Alexander D. Scott

We prove results on the computational complexity of instances of 3SAT in which every variable occurs 3 or 4 times.

more >>>Bruno Codenotti, Amin Saberi, Kasturi Varadarajan, Yinyu Ye

We give a reduction from any two-player game to a special case of

the Leontief exchange economy, with the property that the Nash equilibria of the game and the

equilibria of the market are in one-to-one correspondence.

Our reduction exposes a potential hurdle inherent in solving certain

families of market ...
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Li-Sha Huang, Xiaotie Deng

We consider the computational complexity of the market equilibrium

problem by exploring the structural properties of the Leontief

exchange economy. We prove that, for economies guaranteed to have

a market equilibrium, finding one with maximum social welfare or

maximum individual welfare is NP-hard. In addition, we prove that

counting the ...
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Michael R. Fellows, Frances A. Rosamond, Udi Rotics, Stefan Szeider

Clique-width is a graph parameter, defined by a composition mechanism

for vertex-labeled graphs, which measures in a certain sense the

complexity of a graph. Hard graph problems (e.g., problems

expressible in Monadic Second Order Logic, that includes NP-hard

problems) can be solved efficiently for graphs of certified small

clique-width. It ...
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Michael R. Fellows, Frances A. Rosamond, Udi Rotics, Stefan Szeider

Clique-width is a graph parameter that measures in a certain sense the

complexity of a graph. Hard graph problems (e.g., problems

expressible in Monadic Second Order Logic with second-order

quantification on vertex sets, that includes NP-hard problems) can be

solved efficiently for graphs of certified small clique-width. It is

widely ...
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Jesper Torp Kristensen, Peter Bro Miltersen

We present an efficient reduction mapping undirected graphs

G with n = 2^k vertices for integers k

to tables of partially specified Boolean functions

g: {0,1}^(4k+1) -> {0,1,*} so that for any integer m,

G has a vertex colouring using m colours if and only if g ...
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Christian Glaßer, Christian Reitwießner, Heinz Schmitz, Maximilian Witek

We systematically study the hardness and the approximability of combinatorial multi-objective NP optimization problems (multi-objective problems, for short).

We define solution notions that precisely capture the typical algorithmic tasks in multi-objective optimization. These notions inherit polynomial-time Turing reducibility from multivalued functions, which allows us to compare the solution notions and ... more >>>

Daniele Micciancio

We prove that the Shortest Vector Problem (SVP) on point lattices is NP-hard to approximate for any constant factor under polynomial time reverse unfaithful random reductions. These are probabilistic reductions with one-sided error that produce false negatives with small probability, but are guaranteed not to produce false positives regardless of ... more >>>

Subhash Khot, Madhur Tulsiani, Pratik Worah

A boolean predicate $f:\{0,1\}^k\to\{0,1\}$ is said to be {\em somewhat approximation resistant} if for some constant $\tau > \frac{|f^{-1}(1)|}{2^k}$, given a $\tau$-satisfiable instance of the MAX-$k$-CSP$(f)$ problem, it is NP-hard to find an assignment that {\it strictly beats} the naive algorithm that outputs a uniformly random assignment. Let $\tau(f)$ denote ... more >>>

Daniele Micciancio

The Minimum Distance Problem (MDP), i.e., the computational task of evaluating (exactly or approximately) the minimum distance of a linear code, is a well known NP-hard problem in coding theory. A key element in essentially all known proofs that MDP is NP-hard is the construction of a combinatorial object that ... more >>>

Andrej Bogdanov, Christina Brzuska

We prove that if the hardness of inverting a size-verifiable one-way function can

be based on NP-hardness via a general (adaptive) reduction, then coAM is contained in NP. This

claim was made by Akavia, Goldreich, Goldwasser, and Moshkovitz (STOC 2006), but

was later retracted (STOC 2010).