$k$ -Server的击中集松弛及时间窗的扩展
A Hitting Set Relaxation for $k$-Server and an Extension to Time-Windows
摘要：
研究了带时间窗的$k$ -服务器问题。在这个问题中，每个请求$i$在时间$b_i$到达$n$ -点度量空间的某个点$v_i$，并带有截止期限$e_i$ .必须在间隔$[b_i, e_i]$内的某个时间将其中一个$k$服务器移动到$v_i$，以满足此请求。我们给出了这个问题的一个在线算法的竞争比${\rm polylog} (n,Δ)$，其中$Δ$是度量空间的纵横比。在我们的工作之前，这个问题已知的最佳竞争比是Azar等人给出的$O(k \cdot {\rm polylog}(n))$（STOC 2017）。我们的算法是基于一个新的覆盖线性规划松弛的$k$ -服务器上的HST。该LP自然对应于$k$ -服务器的最小成本流公式，并且容易扩展到时间窗口的情况。我们给出了一个在线算法，获得一个可行的分数的解决方案，这个LP，和会计的解决方案的成本的原始对偶分析框架。结合这两种算法，得到了一个新的具有多对数竞争比的$k$ -服务器算法，并将其推广到时间窗情形。我们的主要技术贡献在于将覆盖LP视为在树的每个内部节点处产生{ \em截断}覆盖LP，这使我们能够考虑子树之间的服务器移动。我们希望这种LP松弛和算法/分析将是解决$k$ -服务器和相关问题的有用工具。
英文摘要：
We study the $k$-server problem with time-windows. In this problem, each request $i$ arrives at some point $v_i$ of an $n$-point metric space at time $b_i$ and comes with a deadline $e_i$. One of the $k$ servers must be moved to $v_i$ at some time in the interval $[b_i, e_i]$ to satisfy this request. We give an online algorithm for this problem with a competitive ratio of ${\rm polylog} (n,Δ)$, where $Δ$ is the aspect ratio of the metric space. Prior to our work, the best competitive ratio known for this problem was $O(k \cdot {\rm polylog}(n))$ given by Azar et al. (STOC 2017). Our algorithm is based on a new covering linear program relaxation for $k$-server on HSTs. This LP naturally corresponds to the min-cost flow formulation of $k$-server, and easily extends to the case of time-windows. We give an online algorithm for obtaining a feasible fractional solution for this LP, and a primal dual analysis framework for accounting the cost of the solution. Together, they yield a new $k$-server algorithm with poly-logarithmic competitive ratio, and extend to the time-windows case as well. Our principal technical contribution lies in thinking of the covering LP as yielding a {\em truncated} covering LP at each internal node of the tree, which allows us to keep account of server movements across subtrees. We hope that this LP relaxation and the algorithm/analysis will be a useful tool for addressing $k$-server and related problems.
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