arXiv subjects

Shi Li

Publications and source records attributed to Shi Li.

At least 73 records · Page 4Linked to original sources

arXiv2021

Online Food Delivery to Minimize Maximum Flow Time

We study a common delivery problem encountered in nowadays online food-ordering platforms: Customers order dishes online, and the restaurant delivers the food after receiving the order. Specifically, we study a problem where $k$ vehicles of capacity $c$ are serving a set of requests ordering food from one restaurant. After a request arrives, it can be served by a vehicle moving from the restaurant to its delivery location. We are interested in serving all requests while minimizing the maximum flow-time, i.e., the maximum time length a customer waits to receive his/her food after submitting the order. We show that the problem is hard in both offline and online settings: There is a hardness of approximation of $Ω(n)$ for the offline problem, and a lower bound of $Ω(n)$ on the competitive ratio of any online algorithm, where $n$ is number of points in the metric. Our main result is an $O(1)$-competitive online algorithm for the uncapaciated (i.e, $c = \infty$) food delivery problem on tree metrics. Then we consider the speed-augmentation model. We develop an exponential time $(1+ε)$-speeding $O(1/ε)$-competitive algorithm for any $ε> 0$. A polynomial time algorithm can be obtained with a speeding factor of $α_{TSP}+ ε$ or $α_{CVRP}+ ε$, depending on whether the problem is uncapacitated. Here $α_{TSP}$ and $α_{CVRP}$ are the best approximation factors for the traveling salesman (TSP) and capacitated vehicle routing (CVRP) problems respectively. We complement the results with some negative ones.

cs.DS↗

arXiv2021

56 GBaud PAM-4 100 km Transmission System with Photonic Processing Schemes

Analog photonic computing has been proposed and tested in recent years as an alternative approach for data recovery in fiber transmission systems. Photonic reservoir computing, performing nonlinear transformations of the transmitted signals and exhibiting internal fading memory, has been found advantageous for this kind of processing. In this work, we show that the effectiveness of the internal fading memory depends significantly on the properties of the signal to be processed. Specifically, we demonstrate two experimental photonic post-processing schemes for a 56 GBaud PAM-4 experimental transmission system, with 100 km uncompensated standard single-mode fiber and direct detection. We show that, for transmission systems with significant chromatic dispersion, the contribution of a photonic reservoir's fading memory to the computational performance is limited. In a comparison between the data recovery performances between a reservoir computing and an extreme learning machine fiber-based configuration, we find that both offer equivalent data recovery. The extreme learning machine approach eliminates the necessity of external recurrent connectivity, which simplifies the system and increases the computation speed. Above 31 dB OSNR, the photonics-based equalization exhibits a lower BER than the respective offline DSP-based KK receiver.

eess.SP↗

arXiv2020

Approximation algorithms for stochastic clustering

We consider stochastic settings for clustering, and develop provably-good approximation algorithms for a number of these notions. These algorithms yield better approximation ratios compared to the usual deterministic clustering setting. Additionally, they offer a number of advantages including clustering which is fairer and has better long-term behavior for each user. In particular, they ensure that *every user* is guaranteed to get good service (on average). We also complement some of these with impossibility results.

cs.DS↗

arXiv2020

Estimating Stochastic Linear Combination of Non-linear Regressions Efficiently and Scalably

Recently, many machine learning and statistical models such as non-linear regressions, the Single Index, Multi-index, Varying Coefficient Index Models and Two-layer Neural Networks can be reduced to or be seen as a special case of a new model which is called the \textit{Stochastic Linear Combination of Non-linear Regressions} model. However, due to the high non-convexity of the problem, there is no previous work study how to estimate the model. In this paper, we provide the first study on how to estimate the model efficiently and scalably. Specifically, we first show that with some mild assumptions, if the variate vector $x$ is multivariate Gaussian, then there is an algorithm whose output vectors have $\ell_2$-norm estimation errors of $O(\sqrt{\frac{p}{n}})$ with high probability, where $p$ is the dimension of $x$ and $n$ is the number of samples. The key idea of the proof is based on an observation motived by the Stein's lemma. Then we extend our result to the case where $x$ is bounded and sub-Gaussian using the zero-bias transformation, which could be seen as a generalization of the classic Stein's lemma. We also show that with some additional assumptions there is an algorithm whose output vectors have $\ell_\infty$-norm estimation errors of $O(\frac{1}{\sqrt{p}}+\sqrt{\frac{p}{n}})$ with high probability. We also provide a concrete example to show that there exists some link function which satisfies the previous assumptions. Finally, for both Gaussian and sub-Gaussian cases we propose a faster sub-sampling based algorithm and show that when the sub-sample sizes are large enough then the estimation errors will not be sacrificed by too much. Experiments for both cases support our theoretical results. To the best of our knowledge, this is the first work that studies and provides theoretical guarantees for the stochastic linear combination of non-linear regressions model.

cs.LG↗

arXiv2020

Robust High Dimensional Expectation Maximization Algorithm via Trimmed Hard Thresholding

In this paper, we study the problem of estimating latent variable models with arbitrarily corrupted samples in high dimensional space ({\em i.e.,} $d\gg n$) where the underlying parameter is assumed to be sparse. Specifically, we propose a method called Trimmed (Gradient) Expectation Maximization which adds a trimming gradients step and a hard thresholding step to the Expectation step (E-step) and the Maximization step (M-step), respectively. We show that under some mild assumptions and with an appropriate initialization, the algorithm is corruption-proofing and converges to the (near) optimal statistical rate geometrically when the fraction of the corrupted samples $ε$ is bounded by $ \tilde{O}(\frac{1}{\sqrt{n}})$. Moreover, we apply our general framework to three canonical models: mixture of Gaussians, mixture of regressions and linear regression with missing covariates. Our theory is supported by thorough numerical results.

stat.ML↗

arXiv2020

Consistent $k$-Median: Simpler, Better and Robust

In this paper we introduce and study the online consistent $k$-clustering with outliers problem, generalizing the non-outlier version of the problem studied in [Lattanzi-Vassilvitskii, ICML17]. We show that a simple local-search based online algorithm can give a bicriteria constant approximation for the problem with $O(k^2 \log^2 (nD))$ swaps of medians (recourse) in total, where $D$ is the diameter of the metric. When restricted to the problem without outliers, our algorithm is simpler, deterministic and gives better approximation ratio and recourse, compared to that of [Lattanzi-Vassilvitskii, ICML17].

cs.DS↗

arXiv2020

Hierarchy-Based Algorithms for Minimizing Makespan under Precedence and Communication Constraints

We consider the classic problem of scheduling jobs with precedence constraints on a set of identical machines to minimize the makespan objective function. Understanding the exact approximability of the problem when the number of machines is a constant is a well-known question in scheduling theory. Indeed, an outstanding open problem from the classic book of Garey and Johnson asks whether this problem is NP-hard even in the case of 3 machines and unit-length jobs. In a recent breakthrough, Levey and Rothvoss gave a $(1+ε)$-approximation algorithm, which runs in nearly quasi-polynomial time, for the case when job have unit lengths. However, a substantially more difficult case where jobs have arbitrary processing lengths has remained open. We make progress on this more general problem. We show that there exists a $(1+ε)$-approximation algorithm (with similar running time as that of Levey and Rothvoss) for the non-migratory setting: when every job has to be scheduled entirely on a single machine, but within a machine the job need not be scheduled during consecutive time steps. Further, we also show that our algorithmic framework generalizes to another classic scenario where, along with the precedence constraints, the jobs also have communication delay constraints. Both of these fundamental problems are highly relevant to the practice of datacenter scheduling.

cs.DS↗

arXiv2020

On Approximating Degree-Bounded Network Design Problems

Directed Steiner Tree (DST) is a central problem in combinatorial optimization and theoretical computer science: Given a directed graph $G=(V, E)$ with edge costs $c \in \mathbb{R}_{\geq 0}^E$, a root $r \in V$ and $k$ terminals $K\subseteq V$, we need to output the minimum-cost arborescence in $G$ that contains an $r$\textrightarrow $t$ path for every $t \in K$. Recently, Grandoni, Laekhanukit and Li, and independently Ghuge and Nagarajan, gave quasi-polynomial time $O(\log^2k/\log \log k)$-approximation algorithms for the problem, which are tight under popular complexity assumptions. In this paper, we consider the more general Degree-Bounded Directed Steiner Tree (DB-DST) problem, where we are additionally given a degree bound $d_v$ on each vertex $v \in V$, and we require that every vertex $v$ in the output tree has at most $d_v$ children. We give a quasi-polynomial time $(O(\log n \log k), O(\log^2 n))$-bicriteria approximation: The algorithm produces a solution with cost at most $O(\log n\log k)$ times the cost of the optimum solution that violates the degree constraints by at most a factor of $O(\log^2n)$. This is the first non-trivial result for the problem. While our cost-guarantee is nearly optimal, the degree violation factor of $O(\log^2n)$ is an $O(\log n)$-factor away from the approximation lower bound of $Ω(\log n)$ from the set-cover hardness. The hardness result holds even on the special case of the {\em Degree-Bounded Group Steiner Tree} problem on trees (DB-GST-T). With the hope of closing the gap, we study the question of whether the degree violation factor can be made tight for this special case. We answer the question in the affirmative by giving an $(O(\log n\log k), O(\log n))$-bicriteria approximation algorithm for DB-GST-T.

cs.DS↗

arXiv2020

Towards PTAS for Precedence Constrained Scheduling via Combinatorial Algorithms

We study the classic problem of scheduling $n$ precedence constrained unit-size jobs on $m = O(1)$ machines so as to minimize the makespan. In a recent breakthrough, Levey and Rothvoss \cite{LR16} developed a $(1+ε)$-approximation for the problem with running time $\exp\Big(\exp\Big(O\big(\frac{m^2}{ε^2}\log^2\log n\big)\Big)\Big)$, via the Sherali-Adams lift of the basic linear programming relaxation for the problem by $\exp\Big(O\big(\frac{m^2}{ε^2}\log^2\log n\big)\Big)$ levels. Garg \cite{Garg18} recently improved the number of levels to $\log ^{O(m^2/ε^2)}n$, and thus the running time to $\exp\big(\log ^{O(m^2/ε^2)}n\big)$, which is quasi-polynomial for constant $m$ and $ε$. In this paper we present an algorithm that achieves $(1+ε)$-approximation for the problem with running time $n^{O\left(\frac{m^4}{ε^3}\log^3\log n\right)}$, which is very close to a polynomial for constant $m$ and $ε$. Unlike the algorithms of Levey-Rothvoss and Garg, which are based on linear-programming hierarchy, our algorithm is purely combinatorial. For this problem, we show that the conditioning operations on the lifted LP solution can be replaced by making guesses about the optimum schedule.

cs.DS↗

arXiv2020

Covering the Relational Join

In this paper, we initiate a theoretical study of what we call the join covering problem. We are given a natural join query instance $Q$ on $n$ attributes and $m$ relations $(R_i)_{i \in [m]}$. Let $J_{Q} = \ \Join_{i=1}^m R_i$ denote the join output of $Q$. In addition to $Q$, we are given a parameter $Δ: 1\le Δ\le n$ and our goal is to compute the smallest subset $\mathcal{T}_{Q, Δ} \subseteq J_{Q}$ such that every tuple in $J_{Q}$ is within Hamming distance $Δ- 1$ from some tuple in $\mathcal{T}_{Q, Δ}$. The join covering problem captures both computing the natural join from database theory and constructing a covering code with covering radius $Δ- 1$ from coding theory, as special cases. We consider the combinatorial version of the join covering problem, where our goal is to determine the worst-case $|\mathcal{T}_{Q, Δ}|$ in terms of the structure of $Q$ and value of $Δ$. One obvious approach to upper bound $|\mathcal{T}_{Q, Δ}|$ is to exploit a distance property (of Hamming distance) from coding theory and combine it with the worst-case bounds on output size of natural joins (AGM bound hereon) due to Atserias, Grohe and Marx [SIAM J. of Computing'13]. Somewhat surprisingly, this approach is not tight even for the case when the input relations have arity at most two. Instead, we show that using the polymatroid degree-based bound of Abo Khamis, Ngo and Suciu [PODS'17] in place of the AGM bound gives us a tight bound (up to constant factors) on the $|\mathcal{T}_{Q, Δ}|$ for the arity two case. We prove lower bounds for $|\mathcal{T}_{Q, Δ}|$ using well-known classes of error-correcting codes e.g, Reed-Solomon codes. We can extend our results for the arity two case to general arity with a polynomial gap between our upper and lower bounds.

cs.DB↗

arXiv2020

Topology Dependent Bounds For FAQs

In this paper, we prove topology dependent bounds on the number of rounds needed to compute Functional Aggregate Queries (FAQs) studied by Abo Khamis et al. [PODS 2016] in a synchronous distributed network under the model considered by Chattopadhyay et al. [FOCS 2014, SODA 2017]. Unlike the recent work on computing database queries in the Massively Parallel Computation model, in the model of Chattopadhyay et al., nodes can communicate only via private point-to-point channels and we are interested in bounds that work over an {\em arbitrary} communication topology. This is the first work to consider more practically motivated problems in this distributed model. For the sake of exposition, we focus on two special problems in this paper: Boolean Conjunctive Query (BCQ) and computing variable/factor marginals in Probabilistic Graphical Models (PGMs). We obtain tight bounds on the number of rounds needed to compute such queries as long as the underlying hypergraph of the query is $O(1)$-degenerate and has $O(1)$-arity. In particular, the $O(1)$-degeneracy condition covers most well-studied queries that are efficiently computable in the centralized computation model like queries with constant treewidth. These tight bounds depend on a new notion of `width' (namely internal-node-width) for Generalized Hypertree Decompositions (GHDs) of acyclic hypergraphs, which minimizes the number of internal nodes in a sub-class of GHDs. To the best of our knowledge, this width has not been studied explicitly in the theoretical database literature. Finally, we consider the problem of computing the product of a vector with a chain of matrices and prove tight bounds on its round complexity (over the finite field of two elements) using a novel min-entropy based argument.

cs.DC↗

arXiv2020

The Power of Recourse: Better Algorithms for Facility Location in Online and Dynamic Models

In this paper we study the facility location problem in the online with recourse and dynamic algorithm models. In the online with recourse model, clients arrive one by one and our algorithm needs to maintain good solutions at all time steps with only a few changes to the previously made decisions (called recourse). We show that the classic local search technique can lead to a $(1+\sqrt{2}+ε)$-competitive online algorithm for facility location with only $O\left(\frac{\log n}ε\log\frac1ε\right)$ amortized facility and client recourse. We then turn to the dynamic algorithm model for the problem, where the main goal is to design fast algorithms that maintain good solutions at all time steps. We show that the result for online facility location, combined with the randomized local search technique of Charikar and Guha [10], leads to an $O(1+\sqrt{2}+ε)$ approximation dynamic algorithm with amortized update time of $\tilde O(n)$ in the incremental setting. Notice that the running time is almost optimal, since in general metric space it takes $Ω(n)$ time to specify a new client's position. The approximation factor of our algorithm also matches the best offline analysis of the classic local search algorithm. Finally, we study the fully dynamic model for facility location, where clients can both arrive and depart. Our main result is an $O(1)$-approximation algorithm in this model with $O(|F|)$ preprocessing time and $O(\log^3 D)$ amortized update time for the HST metric spaces. Using the seminal results of Bartal [4] and Fakcharoenphol, Rao and Talwar [17], which show that any arbitrary $N$-point metric space can be embedded into a distribution over HSTs such that the expected distortion is at most $O(\log N)$, we obtain a $O(\log |F|)$ approximation with preprocessing time of $O(|F|^2\log |F|)$ and $O(\log^3 D)$ amortized update time.

cs.DS↗

arXiv2019

Facility Location Problem in Differential Privacy Model Revisited

In this paper we study the uncapacitated facility location problem in the model of differential privacy (DP) with uniform facility cost. Specifically, we first show that, under the hierarchically well-separated tree (HST) metrics and the super-set output setting that was introduced in Gupta et. al., there is an $ε$-DP algorithm that achieves an $O(\frac{1}ε)$(expected multiplicative) approximation ratio; this implies an $O(\frac{\log n}ε)$ approximation ratio for the general metric case, where $n$ is the size of the input metric. These bounds improve the best-known results given by Gupta et. al. In particular, our approximation ratio for HST-metrics is independent of $n$, and the ratio for general metrics is independent of the aspect ratio of the input metric. On the negative side, we show that the approximation ratio of any $ε$-DP algorithm is lower bounded by $Ω(\frac{1}{\sqrtε})$, even for instances on HST metrics with uniform facility cost, under the super-set output setting. The lower bound shows that the dependence of the approximation ratio for HST metrics on $ε$ can not be removed or greatly improved. Our novel methods and techniques for both the upper and lower bound may find additional applications.

cs.DS↗

arXiv2018

Distributed $k$-Clustering for Data with Heavy Noise

In this paper, we consider the $k$-center/median/means clustering with outliers problems (or the $(k, z)$-center/median/means problems) in the distributed setting. Most previous distributed algorithms have their communication costs linearly depending on $z$, the number of outliers. Recently Guha et al. overcame this dependence issue by considering bi-criteria approximation algorithms that output solutions with $2z$ outliers. For the case where $z$ is large, the extra $z$ outliers discarded by the algorithms might be too large, considering that the data gathering process might be costly. In this paper, we improve the number of outliers to the best possible $(1+ε)z$, while maintaining the $O(1)$-approximation ratio and independence of communication cost on $z$. The problems we consider include the $(k, z)$-center problem, and $(k, z)$-median/means problems in Euclidean metrics. Implementation of the our algorithm for $(k, z)$-center shows that it outperforms many previous algorithms, both in terms of the communication cost and quality of the output solution.

cs.DC↗

arXiv2018

$O(\log^2k/\log\log{k})$-Approximation Algorithm for Directed Steiner Tree: A Tight Quasi-Polynomial-Time Algorithm

In the Directed Steiner Tree (DST) problem we are given an $n$-vertex directed edge-weighted graph, a root $r$, and a collection of $k$ terminal nodes. Our goal is to find a minimum-cost arborescence that contains a directed path from $r$ to every terminal. We present an $O(\log^2 k/\log\log{k})$-approximation algorithm for DST that runs in quasi-polynomial-time. By adjusting the parameters in the hardness result of Halperin and Krauthgamer, we show the matching lower bound of $Ω(\log^2{k}/\log\log{k})$ for the class of quasi-polynomial-time algorithms. This is the first improvement on the DST problem since the classical quasi-polynomial-time $O(\log^3 k)$ approximation algorithm by Charikar et al. (The paper erroneously claims an $O(\log^2k)$ approximation due to a mistake in prior work.) Our approach is based on two main ingredients. First, we derive an approximation preserving reduction to the Label-Consistent Subtree (LCST) problem. The LCST instance has quasi-polynomial size and logarithmic height. We remark that, in contrast, Zelikovsky's heigh-reduction theorem used in all prior work on DST achieves a reduction to a tree instance of the related Group Steiner Tree (GST) problem of similar height, however losing a logarithmic factor in the approximation ratio. Our second ingredient is an LP-rounding algorithm to approximately solve LCST instances, which is inspired by the framework developed by Rothvoß. We consider a Sherali-Adams lifting of a proper LP relaxation of LCST. Our rounding algorithm proceeds level by level from the root to the leaves, rounding and conditioning each time on a proper subset of label variables. A small enough (namely, polylogarithmic) number of Sherali-Adams lifting levels is sufficient to condition up to the leaves.

cs.DS↗

arXiv2018

A Polynomial Time Constant Approximation For Minimizing Total Weighted Flow-time

We consider the classic scheduling problem of minimizing the total weighted flow-time on a single machine (min-WPFT), when preemption is allowed. In this problem, we are given a set of $n$ jobs, each job having a release time $r_j$, a processing time $p_j$, and a weight $w_j$. The flow-time of a job is defined as the amount of time the job spends in the system before it completes; that is, $F_j = C_j - r_j$, where $C_j$ is the completion time of job. The objective is to minimize the total weighted flow-time of jobs. This NP-hard problem has been studied quite extensively for decades. In a recent breakthrough, Batra, Garg, and Kumar presented a {\em pseudo-polynomial} time algorithm that has an $O(1)$ approximation ratio. The design of a truly polynomial time algorithm, however, remained an open problem. In this paper, we show a transformation from pseudo-polynomial time algorithms to polynomial time algorithms in the context of min-WPFT. Our result combined with the result of Batra, Garg, and Kumar settles the long standing conjecture that there is a polynomial time algorithm with $O(1)$-approximation for min-WPFT.

cs.DS↗

arXiv2018

Flow-time Optimization For Concurrent Open-Shop and Precedence Constrained Scheduling Models

Scheduling a set of jobs over a collection of machines is a fundamental problem that needs to be solved millions of times a day in various computing platforms: in operating systems, in large data clusters, and in data centers. Along with makespan, flow-time, which measures the length of time a job spends in a system before it completes, is arguably the most important metric to measure the performance of a scheduling algorithm. In recent years, there has been a remarkable progress in understanding flow-time based objective functions in diverse settings such as unrelated machines scheduling, broadcast scheduling, multi-dimensional scheduling, to name a few. Yet, our understanding of the flow-time objective is limited mostly to the scenarios where jobs have simple structures; in particular, each job is a single self contained entity. On the other hand, in almost all real world applications, think of MapReduce settings for example, jobs have more complex structures. In this paper, we consider two classical scheduling models that capture complex job structures: 1) concurrent open-shop scheduling and 2) precedence constrained scheduling. Our main motivation to study these problems specifically comes from their relevance to two scheduling problems that have gained importance in the context of data centers: co-flow scheduling and DAG scheduling. We design almost optimal approximation algorithms for open-shop scheduling and precedence constrained scheduling, and show hardness results.

cs.DS↗

arXiv2018

The Design of Data Acquisition System for EAST Technical Diagnostic System

EAST (Experimental Advanced Superconducting Tokamak) Technical Diagnostic System (TDS) is used to monitor the outlet temperature of all superconducting coils, in case of temperature anomaly, it will trigger safety interlock system to meet EAST device safety. The data acquisition system of TDS is in charge of continuous data acquisition of the nitrogen and helium temperature signals, TDS security alarm and long-term data storage. It supports continuous data acquisition and pulse data acquisition. The data acquisition of the nitrogen temperature signals is based on the PXI technology while obtaining the helium temperature signals from Lake Shore model 224 temperature monitors with VISA standard. After data conversion, all the data will be stored in MySQL and MDSPlus for long-term storage. It should output TDS fault signal and status signal to trigger the safety interlock system to take actions after threshold evaluation of key temperature signals. It publishes part of real-time TDS data to the cryogenic system and provides an information inquiry service to the TDS administrator. The system has been used in 2018 EAST campaign.

physics.ins-det↗