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

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

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

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