17,506 research outputs found

### On Unique Games with Negative Weights

In this paper, the author defines Generalized Unique Game Problem (GUGP),
where weights of the edges are allowed to be negative. Two special types of
GUGP are illuminated, GUGP-NWA, where the weights of all edges are negative,
and GUGP-PWT($\rho$), where the total weight of all edges are positive and the
negative-positive ratio is at most $\rho$. The author investigates the
counterpart of the Unique Game Conjecture on GUGP-PWT($\rho$). The author shows
that Unique Game Conjecture on GUGP-PWT(1) holds true, and Unique Game
Conjecture on GUGP-PWT(1/2) holds true, if the 2-to-1 Conjecture holds true.
The author poses an open problem whether Unique Game Conjecture holds true on
GUGP-PWT($\rho$) with $0<\rho<1$.Comment: 7 pages, accepted by COCOA 201

### A Tighter Analysis of Setcover Greedy Algorithm for Test Set

Setcover greedy algorithm is a natural approximation algorithm for test set
problem. This paper gives a precise and tighter analysis of performance
guarantee of this algorithm. The author improves the performance guarantee
$2\ln n$ which derives from set cover problem to $1.1354\ln n$ by applying the
potential function technique. In addition, the author gives a nontrivial lower
bound $1.0004609\ln n$ of performance guarantee of this algorithm. This lower
bound, together with the matching bound of information content heuristic,
confirms the fact information content heuristic is slightly better than
setcover greedy algorithm in worst case.Comment: 12 pages, 3 figures, Revised versio

### Improved Approximability Result for Test Set with Small Redundancy

Test set with redundancy is one of the focuses in recent bioinformatics
research. Set cover greedy algorithm (SGA for short) is a commonly used
algorithm for test set with redundancy. This paper proves that the
approximation ratio of SGA can be $(2-\frac{1}{2r})\ln n+{3/2}\ln r+O(\ln\ln
n)$ by using the potential function technique. This result is better than the
approximation ratio $2\ln n$ which directly derives from set multicover, when
$r=o(\frac{\ln n}{\ln\ln n})$, and is an extension of the approximability
results for plain test set.Comment: 7 page

### Approximation Resistance by Disguising Biased Distributions

In this short note, the author shows that the gap problem of some 3-XOR is
NP-hard and can be solved by running Charikar\&Wirth's SDP algorithm for two
rounds. To conclude, the author proves that $P=NP$.Comment: 6 pages, short not

### Refuting Unique Game Conjecture

In this short note, the author shows that the gap problem of some $k$-CSPs
with the support of its predicate the ground of a balanced pairwise independent
distribution can be solved by a modified version of Hast's Algorithm BiLin that
calls Charikar\&Wirth's SDP algorithm for two rounds in polynomial time, when
$k$ is sufficiently large, the support of its predicate is combined by the
grounds of three biased homogeneous distributions and the three biases satisfy
certain conditions. To conclude, the author refutes Unique Game Conjecture,
assuming $P\ne NP$.Comment: 6 pages, short note. arXiv admin note: substantial text overlap with
arXiv:1401.652

### Strengthened Hardness for Approximating Minimum Unique Game and Small Set Expansion

In this paper, the author puts forward a variation of Feige's Hypothesis,
which claims that it is hard on average refuting Unbalanced Max 3-XOR under
biased assignments on a natural distribution. Under this hypothesis, the author
strengthens the previous known hardness for approximating Minimum Unique Game,
$5/4-\epsilon$, by proving that Min 2-Lin-2 is hard to within $3/2-\epsilon$
and strengthens the previous known hardness for approximating Small Set
Expansion, $4/3-\epsilon$, by proving that Min Bisection is hard to approximate
within $3-\epsilon$. In addition, the author discusses the limitation of this
method to show that it can strengthen the hardness for approximating Minimum
Unique Game to $2-\kappa$ where $\kappa$ is a small absolute positive, but is
short of proving $\omega_k(1)$ hardness for Minimum Unique Game (or Small Set
Expansion), by assuming a generalization of this hypothesis on Unbalanced Max
k-CSP with Samorodnitsky-Trevisan hypergraph predicate.Comment: 11 pages, 1 figur

### On the cycles of components of disconnected Julia sets

For any integers $d\ge 3$ and $n\ge 1$, we construct a hyperbolic rational
map of degree $d$ such that it has $n$ cycles of the connected components of
its Julia set except single points and Jordan curves.Comment: 30 pages, 9 figure

### Renormalization and wandering continua of rational maps

Renormalizations can be considered as building blocks of complex dynamical
systems. This phenomenon has been widely studied for iterations of polynomials
of one complex variable. Concerning non-polynomial hyperbolic rational maps, a
recent work of Cui-Tan shows that these maps can be decomposed into
postcritically finite renormalization pieces. The main purpose of the present
work is to perform the surgery one step deeper. Based on Thurston's idea of
decompositions along multicurves, we introduce a key notion of Cantor
multicurves (a stable multicurve generating infinitely many homotopic curves
under pullback), and prove that any postcritically finite piece having a Cantor
multicurve can be further decomposed into smaller postcritically finite
renormalization pieces. As a byproduct, we establish the presence of separating
wandering continua in the corresponding Julia sets. Contrary to the polynomial
case, we exploit tools beyond the category of analytic and quasiconformal maps,
such as Rees-Shishikura's semi-conjugacy for topological branched coverings
that are Thurston-equivalent to rational maps.Comment: 24 pages, 2 figures. This paper has been withdrawn by the author
since it is the old version of the paper http://arxiv.org/abs/1403.502

### Renormalizations and wandering Jordan curves of rational maps

We realize a dynamical decomposition for a post-critically finite rational
map which admits a combinatorial decomposition. We split the Riemann sphere
into two completely invariant subsets. One is a subset of the Julia set
consisting of uncountably many Jordan curve components. Most of them are
wandering. The other consists of components that are pullbacks of finitely many
renormalizations, together with possibly uncountably many points. The quotient
action on the decomposed pieces is encoded by a dendrite dynamical system. We
also introduce a surgery procedure to produce post-critically finite rational
maps with wandering Jordan curves and prescribed renormalizations.Comment: 49 pages, 3 figure

### Non-convex Fraction Function Penalty: Sparse Signals Recovered from Quasi-linear Systems

The goal of compressed sensing is to reconstruct a sparse signal under a few
linear measurements far less than the dimension of the ambient space of the
signal. However, many real-life applications in physics and biomedical sciences
carry some strongly nonlinear structures, and the linear model is no longer
suitable. Compared with the compressed sensing under the linear circumstance,
this nonlinear compressed sensing is much more difficult, in fact also NP-hard,
combinatorial problem, because of the discrete and discontinuous nature of the
$\ell_{0}$-norm and the nonlinearity. In order to get a convenience for sparse
signal recovery, we set most of the nonlinear models have a smooth quasi-linear
nature in this paper, and study a non-convex fraction function $\rho_{a}$ in
this quasi-linear compressed sensing. We propose an iterative fraction
thresholding algorithm to solve the regularization problem $(QP_{a}^{\lambda})$
for all $a>0$. With the change of parameter $a>0$, our algorithm could get a
promising result, which is one of the advantages for our algorithm compared
with other algorithms. Numerical experiments show that our method performs much
better compared with some state-of-art methods

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