Randomized k-server algorithms for growth-rate bounded graphs

Fusible HSTs and the randomized k-server conjecture

We exhibit an $O((\log k)^6)$-competitive randomized algorithm for the $k$-server problem on any metric space. It is shown that a potential-based algorithm for the fractional $k$-server problem on hierarchically separated trees (HSTs) with competitive ratio $f(k)$ can be used to obtain a randomized algorithm for any metric space with competitive ratio $f(k)^2 O((\log k)^2)$. Employing the $O((\log k)^2)$-competitive algorithm for HSTs from our joint work with Bubeck, Cohen, Lee, and Mądry (2017) yields the claimed bound. The best previous result independent of the geometry of the underlying me…

A Polylogarithmic-Competitive Algorithm for the k-Server Problem

We give the first polylogarithmic-competitive randomized online algorithm for the $k$-server problem on an arbitrary finite metric space. In particular, our algorithm achieves a competitive ratio of O(log^3 n log^2 k log log n) for any metric space on n points. Our algorithm improves upon the deterministic (2k-1)-competitive algorithm of Koutsoupias and Papadimitriou [J.ACM'95] whenever n is sub-exponential in k.

随机化 k-服务器猜想是错误的！

The Randomized $k$-Server Conjecture is False!

本文通过构造新的度量空间，证明了随机化 k-服务器问题的竞争比下界为 Ω(log² k)，从而反驳了长期存在的民间猜想，并据此改进了度量任务系统及相关问题的竞争比下界。

Metrical Service Systems with Multiple Servers

We study the problem of metrical service systems with multiple servers (MSSMS), which generalizes two well-known problems -- the $k$-server problem, and metrical service systems. The MSSMS problem is to service requests, each of which is an $l$-point subset of a metric space, using $k$ servers, with the objective of minimizing the total distance traveled by the servers. Feuerstein initiated a study of this problem by proving upper and lower bounds on the deterministic competitive ratio for uniform metric spaces. We improve Feuerstein's analysis of the upper bound and prove that his algorithm a…

k-server via multiscale entropic regularization

We present an $O((\log k)^2)$-competitive randomized algorithm for the $k$-server problem on hierarchically separated trees (HSTs). This is the first $o(k)$-competitive randomized algorithm for which the competitive ratio is independent of the size of the underlying HST. Our algorithm is designed in the framework of online mirror descent where the mirror map is a multiscale entropy. When combined with Bartal's static HST embedding reduction, this leads to an $O((\log k)^2 \log n)$-competitive algorithm on any $n$-point metric space. We give a new dynamic HST embedding that yields an $O((\log k…

加权$k$-服务器问题的一种分解方法

A Decomposition Approach to the Weighted $k$-server Problem

针对均匀度量下加权k-服务器问题竞争比上下界的指数级差距，提出分解方法，将原问题拆为两个松弛版本并证明组合竞争比，给出WkS-RSP的2^{O(k²)}竞争随机算法并证明其Ω(2ᵏ)下界，为设计多项式指数级竞争算法提供路径。

Unbounded lower bound for k-server against weak adversaries

We study the resource augmented version of the $k$-server problem, also known as the $k$-server problem against weak adversaries or the $(h,k)$-server problem. In this setting, an online algorithm using $k$ servers is compared to an offline algorithm using $h$ servers, where $h\le k$. For uniform metrics, it has been known since the seminal work of Sleator and Tarjan (1985) that for any $ε>0$, the competitive ratio drops to a constant if $k=(1+ε) \cdot h$. This result was later generalized to weighted stars (Young 1994) and trees of bounded depth (Bansal et al. 2017). The main open problem for…

Randomized online computation with high probability guarantees

We study the relationship between the competitive ratio and the tail distribution of randomized online minimization problems. To this end, we define a broad class of online problems that includes some of the well-studied problems like paging, k-server and metrical task systems on finite metrics, and show that for these problems it is possible to obtain, given an algorithm with constant expected competitive ratio, another algorithm that achieves the same solution quality up to an arbitrarily small constant error a with high probability; the "high probability" statement is in terms of the optima…

Competitive Algorithms for Generalized k-Server in Uniform Metrics

The generalized k-server problem is a far-reaching extension of the k-server problem with several applications. Here, each server $s_i$ lies in its own metric space $M_i$. A request is a k-tuple $r = (r_1,r_2,\dotsc,r_k)$ and to serve it, we need to move some server $s_i$ to the point $r_i \in M_i$, and the goal is to minimize the total distance traveled by the servers. Despite much work, no f(k)-competitive algorithm is known for the problem for k > 2 servers, even for special cases such as uniform metrics and lines. Here, we consider the problem in uniform metrics and give the first f(k)-com…

Tight Bounds for Online Graph Partitioning

We consider the following online optimization problem. We are given a graph $G$ and each vertex of the graph is assigned to one of $\ell$ servers, where servers have capacity $k$ and we assume that the graph has $\ell \cdot k$ vertices. Initially, $G$ does not contain any edges and then the edges of $G$ are revealed one-by-one. The goal is to design an online algorithm $\operatorname{ONL}$, which always places the connected components induced by the revealed edges on the same server and never exceeds the server capacities by more than $\varepsilon k$ for constant $\varepsilon>0$. Whenever $\op…

面向环需求的动态均衡图划分的多对数竞争算法

Polylog-Competitive Algorithms for Dynamic Balanced Graph Partitioning for Ring Demands

研究面向环通信模式的在线均衡图划分问题，提出 O(log^3 n) 竞争的随机在线算法，并给出相对静态最优为 O(log^2 n) 竞争的另一算法。

时间最优的 $k$-服务器问题

Time-Optimal $k$-Server

本文研究时间最优的k-服务器问题，证明了确定性算法竞争比下界为2k-1，并给出欧几里得空间、均匀度量空间及随机化算法的下界，其中均匀空间下界k可由确定性算法匹配，旨在开启该自然但被忽视问题的系统研究。

Algorithms for Energy Conservation in Heterogeneous Data Centers

Power consumption is the major cost factor in data centers. It can be reduced by dynamically right-sizing the data center according to the currently arriving jobs. If there is a long period with low load, servers can be powered down to save energy. For identical machines, the problem has already been solved optimally by Lin et al. (2013) and Albers and Quedenfeld (2018). In this paper, we study how a data-center with heterogeneous servers can dynamically be right-sized to minimize the energy consumption. There are $d$ different server types with various operating and switching costs. We presen…

带随机奖励的在线$b$-匹配

Online $b$-Matching with Stochastic Rewards

针对带随机奖励的$b$-匹配问题，开发了确定性和随机在线算法竞争比的紧上下界，证明随机算法竞争比不超$1-1/e$，且确定性\textsc{StochasticBalance}算法在非消失概率下可达该竞争比。

加权$k$-服务器问题的高效算法与难解性结果

Efficient Algorithms and Hardness Results for the Weighted $k$-Server Problem

本文研究均匀度量下的加权k-服务器问题，在离线场景证明了该问题的计算下界并给出带资源扩增的LP舍入常数近似算法，在线场景提出两阶段算法将竞争比指数级降低。

A Collection of Lower Bounds for Online Matching on the Line

In the online matching on the line problem, the task is to match a set of requests $R$ online to a given set of servers $S$. The distance metric between any two points in $R\,\cup\, S$ is a line metric and the objective for the online algorithm is to minimize the sum of distances between matched server-request pairs. This problem is well-studied and - despite recent improvements - there is still a large gap between the best known lower and upper bounds: The best known deterministic algorithm for the problem is $O(\log^2n)$-competitive, while the best known deterministic lower bound is $9.001$.…

k-服务器问题中的公平性

Fairness in the k-Server Problem

本文研究了k-服务器问题中的公平性，提出了一种在不损失竞争力的前提下实现公平分配的算法，并探讨了在不同设置下公平性的实现可能性。

Optimal Algorithms for Right-Sizing Data Centers- Extended Version

Electricity cost is a dominant and rapidly growing expense in data centers. Unfortunately, much of the consumed energy is wasted because servers are idle for extended periods of time. We study a capacity management problem that dynamically right-sizes a data center, matching the number of active servers with the varying demand for computing capacity. We resort to a data-center optimization problem introduced by Lin, Wierman, Andrew and Thereska that, over a time horizon, minimizes a combined objective function consisting of operating cost, modeled by a sequence of convex functions, and server …

在云中为多参数作业租赁服务器

Renting Servers for Multi-Parameter Jobs in the Cloud

研究多维云服务器租赁问题，提出单调AF算法类并证明其紧致竞争比，同时设计贪心算法Greedy，在实验中表现优于现有算法。

通过部分差分隐私实现对抗性鲁棒的分布式计数跟踪

Adversarially Robust Distributed Count Tracking via Partial Differential Privacy

本文研究自适应对手下的分布式计数跟踪问题，提出部分差分隐私框架并证明新的泛化定理，从而设计出具有最优通信的鲁棒算法，证明随机化算法的√k优势来自随机性本身而非遗忘型对手假设。

直线上带延迟的在线确定性最小成本二部匹配

Online Deterministic Minimum Cost Bipartite Matching with Delays on a Line

针对实数直线上度量空间未知的带延迟在线最小成本二部完美匹配，本文扩展RM算法，按最小t-净成本设定等待阈值，实现确定性\ ilde{O}(m^{0.5})竞争比。

Metrical task systems on trees via mirror descent and unfair gluing

We consider metrical task systems on tree metrics, and present an $O(\mathrm{depth} \times \log n)$-competitive randomized algorithm based on the mirror descent framework introduced in our prior work on the $k$-server problem. For the special case of hierarchically separated trees (HSTs), we use mirror descent to refine the standard approach based on gluing unfair metrical task systems. This yields an $O(\log n)$-competitive algorithm for HSTs, thus removing an extraneous $\log\log n$ in the bound of Fiat and Mendel (2003). Combined with well-known HST embedding theorems, this also gives an $O…

多服务器作业调度并非易事

Scheduling Multi-Server Jobs is Not Easy

本文研究多服务器作业在线调度，证明等大小作业下任何算法竞争比至少为Ω(K)，并提出竞争比至多K+1的算法、资源增广下竞争比为1的简单算法以及不等大小作业下竞争比至多2K log(K w_max)的算法。