arXiv subjects

Yao Xu

Publications and source records attributed to Yao Xu.

66 records · Page 4Linked to original sources

arXiv2020

Convergence Rates in Almost-Periodic Homogenization of Higher-order Elliptic Systems

This paper concentrates on the quantitative homogenization of higher-order elliptic systems with almost-periodic coefficients in bounded Lipschitz domains. For coefficients which are almost-periodic in the sense of H. Weyl, we establish uniform ocal $L^2$ estimates for the approximate correctors. Under an additional assumption on the frequencies of the coefficients (see (1.10)), we derive the existence of the true correctors as well as the sharp $O(\varepsilon)$ convergence rate in $H^{m-1}$. As a byproduct, the large-scale Hölder estimate and a Liouville theorem are obtained for higher-order elliptic systems with almost-periodic coefficients in the sense of Besicovish. Since (1.10) is not well-defined for the equivalence classes of almost-periodic functions in the sense of H. Weyl or Besicovish, we provide another condition that implies the sharp convergence rate in terms of perturbations on the coefficients.

math.AP↗

arXiv2019

Approximation algorithms for maximally balanced connected graph partition

Given a simple connected graph $G = (V, E)$, we seek to partition the vertex set $V$ into $k$ non-empty parts such that the subgraph induced by each part is connected, and the partition is maximally balanced in the way that the maximum cardinality of these $k$ parts is minimized. We refer this problem to as {\em min-max balanced connected graph partition} into $k$ parts and denote it as {\sc $k$-BGP}. The general vertex-weighted version of this problem on trees has been studied since about four decades ago, which admits a linear time exact algorithm; the vertex-weighted {\sc $2$-BGP} and {\sc $3$-BGP} admit a $5/4$-approximation and a $3/2$-approximation, respectively; but no approximability result exists for {\sc $k$-BGP} when $k \ge 4$, except a trivial $k$-approximation. In this paper, we present another $3/2$-approximation for our cardinality {\sc $3$-BGP} and then extend it to become a $k/2$-approximation for {\sc $k$-BGP}, for any constant $k \ge 3$. Furthermore, for {\sc $4$-BGP}, we propose an improved $24/13$-approximation. To these purposes, we have designed several local improvement operations, which could be useful for related graph partition problems.

cs.DS↗

arXiv2019

Quantitative Estimates in Reiterated Homogenization

This paper investigates quantitative estimates in the homogenization of second-order elliptic systems with periodic coefficients that oscillate on multiple separated scales. We establish large-scale interior and boundary Lipschitz estimates down to the finest microscopic scale via iteration and rescaling arguments. We also obtain a convergence rate in the $L^2$ space by the reiterated homogenization method.

math.AP↗

arXiv2019

Task-Driven Common Representation Learning via Bridge Neural Network

This paper introduces a novel deep learning based method, named bridge neural network (BNN) to dig the potential relationship between two given data sources task by task. The proposed approach employs two convolutional neural networks that project the two data sources into a feature space to learn the desired common representation required by the specific task. The training objective with artificial negative samples is introduced with the ability of mini-batch training and it's asymptotically equivalent to maximizing the total correlation of the two data sources, which is verified by the theoretical analysis. The experiments on the tasks, including pair matching, canonical correlation analysis, transfer learning, and reconstruction demonstrate the state-of-the-art performance of BNN, which may provide new insights into the aspect of common representation learning.

cs.LG↗

arXiv2018

A local search $4/3$-approximation algorithm for the minimum $3$-path partition problem

Given a graph $G = (V, E)$, the $3$-path partition problem is to find a minimum collection of vertex-disjoint paths each of order at most $3$ to cover all the vertices of $V$. It is different from but closely related to the well-known $3$-set cover problem. The best known approximation algorithm for the $3$-path partition problem was proposed recently and has a ratio $13/9$. Here we present a local search algorithm and show, by an amortized analysis, that it is a $4/3$-approximation. This ratio matches up to the best approximation ratio for the $3$-set cover problem.

cs.DS↗

arXiv2018

Convergence rates in homogenization of higher order parabolic systems

This paper is concerned with the optimal convergence rate in homogenization of higher order parabolic systems with bounded measurable, rapidly oscillating periodic coefficients. The sharp $O(\va)$ convergence rate in the space $L^2(0,T; H^{m-1}(\Om))$ is obtained for both the initial-Dirichlet problem and the initial-Neumann problem. The duality argument inspired by \cite{suslinaD2013} is used here.

math.AP↗

arXiv2017

Uniform Boundary Estimates in Homogenization of Higher Order Elliptic Systems

This paper focuses on the uniform boundary estimates in homogenization of a family of higher order elliptic operators $\mathcal{L}_ε$, with rapidly oscillating periodic coefficients. We derive uniform boundary $C^{m-1,λ} (0\!<\!λ\!<\!1)$, $ W^{m,p}$ estimates in $C^1$ domains, as well as uniform boundary $C^{m-1,1}$ estimate in $C^{1,θ} (0\!<\!θ\!<\!1)$ domains without the symmetry assumption on the operator. The proof, motivated by the profound work "S.N. Armstrong and C.~K. Smart, Ann. Sci. Éc. Norm. Supér. (2016), Z. Shen, Anal. PDE (2017)", is based on a suboptimal convergence rate in $H^{m-1}(Ω)$. Compared to "C.E. Kenig, F. Lin and Z. Shen, Arch. Ration. Mech. Anal. (2012), Z. Shen, Anal. PDE (2017)", the convergence rate obtained here does not require the symmetry assumption on the operator, nor additional assumptions on the regularity of $u_0$ (the solution to the homogenized problem), and thus might be of some independent interests even for second order elliptic systems.

math.AP↗

arXiv2017

A $(1.4 + ε)$-approximation algorithm for the $2$-Max-Duo problem

The maximum duo-preservation string mapping (Max-Duo) problem is the complement of the well studied minimum common string partition (MCSP) problem, both of which have applications in many fields including text compression and bioinformatics. $k$-Max-Duo is the restricted version of Max-Duo, where every letter of the alphabet occurs at most $k$ times in each of the strings, which is readily reduced into the well known maximum independent set (MIS) problem on a graph of maximum degree $Δ\le 6(k-1)$. In particular, $2$-Max-Duo can then be approximated arbitrarily close to $1.8$ using the state-of-the-art approximation algorithm for the MIS problem. $2$-Max-Duo was proved APX-hard and very recently a $(1.6 + ε)$-approximation was claimed, for any $ε> 0$. In this paper, we present a vertex-degree reduction technique, based on which, we show that $2$-Max-Duo can be approximated arbitrarily close to $1.4$.

cs.DS↗

arXiv2017

Convergence Rates and Interior Estimates in Homogenization of Higher Order Elliptic Systems

This paper is concerned with the quantitative homogenization of $2m$-order elliptic systems with bounded measurable, rapidly oscillating periodic coefficients. We establish the sharp $O(\varepsilon)$ convergence rate in $W^{m-1, p_0}$ with $p_0=\frac{2d}{d-1}$ in a bounded Lipschitz domain in $\mathbb{R}^d$ as well as the uniform large-scale interior $C^{m-1, 1}$ estimate. With additional smoothness assumptions, the uniform interior $C^{m-1, 1}$, $W^{m,p}$ and $C^{m-1, α}$ estimates are also obtained. As applications of the regularity estimates, we establish asymptotic expansions for fundamental solutions.

math.AP↗

arXiv2017

A local search 2.917-approximation algorithm for duo-preservation string mapping

We study the {\em maximum duo-preservation string mapping} ({\sc Max-Duo}) problem, which is the complement of the well studied {\em minimum common string partition} ({\sc MCSP}) problem. Both problems have applications in many fields including text compression and bioinformatics. Motivated by an earlier local search algorithm, we present an improved approximation and show that its performance ratio is no greater than ${35}/{12} < 2.917$. This beats the current best $3.25$-approximation for {\sc Max-Duo}. The performance analysis of our algorithm is done through a complex yet interesting amortization. Two lower bounds on the locality gap of our algorithm are also provided.

cs.DS↗

arXiv2017

Approximation algorithms for the vertex happiness

We investigate the maximum happy vertices (MHV) problem and its complement, the minimum unhappy vertices (MUHV) problem. We first show that the MHV and MUHV problems are a special case of the supermodular and submodular multi-labeling (Sup-ML and Sub-ML) problems, respectively, by re-writing the objective functions as set functions. The convex relaxation on the Lovász extension, originally presented for the submodular multi-partitioning (Sub-MP) problem, can be extended for the Sub-ML problem, thereby proving that the Sub-ML (Sup-ML, respectively) can be approximated within a factor of $2 - \frac{2}{k}$ ($\frac{2}{k}$, respectively). These general results imply that the MHV and the MUHV problems can also be approximated within $\frac{2}{k}$ and $2 - \frac{2}{k}$, respectively, using the same approximation algorithms. For MHV, this $\frac{2}{k}$-approximation algorithm improves the previous best approximation ratio $\max \{\frac{1}{k}, \frac{1}{Δ+ 1}\}$, where $Δ$ is the maximum vertex degree of the input graph. We also show that an existing LP relaxation is the same as the concave relaxation on the Lovász extension for the Sup-ML problem; we then prove an upper bound of $\frac{2}{k}$ on the integrality gap of the LP relaxation. These suggest that the $\frac{2}{k}$-approximation algorithm is the best possible based on the LP relaxation. For MUHV, we formulate a novel LP relaxation and prove that it is the same as the convex relaxation on the Lovász extension for the Sub-ML problem; we then show a lower bound of $2 - \frac{2}{k}$ on the integrality gap of the LP relaxation. Similarly, these suggest that the $(2 - \frac{2}{k})$-approximation algorithm is the best possible based on the LP relaxation. Lastly, we prove that this $(2 - \frac{2}{k})$-approximation is optimal for the MUHV problem, assuming the Unique Games Conjecture.

cs.DS↗

arXiv2016

Single machine scheduling with job-dependent machine deterioration

We consider the single machine scheduling problem with job-dependent machine deterioration. In the problem, we are given a single machine with an initial non-negative maintenance level, and a set of jobs each with a non-preemptive processing time and a machine deterioration. Such a machine deterioration quantifies the decrement in the machine maintenance level after processing the job. To avoid machine breakdown, one should guarantee a non-negative maintenance level at any time point; and whenever necessary, a maintenance activity must be allocated for restoring the machine maintenance level. The goal of the problem is to schedule the jobs and the maintenance activities such that the total completion time of jobs is minimized. There are two variants of maintenance activities: in the partial maintenance case each activity can be allocated to increase the machine maintenance level to any level not exceeding the maximum; in the full maintenance case every activity must be allocated to increase the machine maintenance level to the maximum. In a recent work, the problem in the full maintenance case has been proven NP-hard; several special cases of the problem in the partial maintenance case were shown solvable in polynomial time, but the complexity of the general problem is left open. In this paper we first prove that the problem in the partial maintenance case is NP-hard, thus settling the open problem; we then design a $2$-approximation algorithm.

cs.DS↗