We consider the k-Service Assignment problem ($$k$$k-SA). The input consists of a network that contains servers and clients. Associated with each client is a demand and a profit. In addition, each client c has a service requirement, where $$\kappa (c)$$κ(c) is a positive integer. A client c is satisfied only if its demand is handled by exactly $$\kappa (c)$$κ(c) neighboring servers. The objective is to maximize the total profit of satisfied clients, while obeying the given capacity limits of the servers. We focus here on the more challenging case of hard constraints, where no profit is granted for partially satisfied clients. This models, e.g., when a client wants, for reasons of fault tolerance, a file to be stored at $$\kappa (c)$$κ(c) or more nearby servers. Other motivations from the literature include resource allocation in 4G cellular networks and machine scheduling on related machines with assignment restrictions. In the r-restricted version of $$k$$k-SA, no client requires more than an r-fraction of the capacity of any adjacent server. We present a (centralized) polynomial-time -approximation algorithm for r-restricted $$k$$k-SA. A variant of this algorithm achieves an approximation ratio of when given a resource augmentation factor of $$1+r$$1+r. We use the latter result to present a -approximation algorithm for $$k$$k-SA. In the distributed setting, we present: (i) a -approximation algorithm for r-restricted $$k$$k-SA, (ii) a -approximation algorithm that uses a resource augmentation factor of $$1+r$$1+r for r-restricted $$k$$k-SA, both for any constant $$\varepsilon >0$$ε>0, and (iii) an -approximation algorithm for $$k$$k-SA (in expectation). The three distributed algorithms run in $$O(k^2 \varepsilon ^{-2} \log ^3 n)$$O(k2ε-2log3n) synchronous rounds (with high probability). In particular, this yields the first distributed -approximation of $$1$$1-SA.

中文翻译：

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k-服务分配的分布式近似

我们考虑 k-服务分配问题 ($$k$$k-SA)。输入由包含服务器和客户端的网络组成。与每个客户相关的是需求和利润。此外，每个客户端 c 都有一个服务需求，其中 $$\kappa (c)$$κ(c) 是一个正整数。仅当客户端 c 的需求恰好由 $$\kappa (c)$$κ(c) 相邻服务器处理时，客户端 c 才会得到满足。目标是最大化满意客户的总利润，同时遵守服务器的给定容量限制。我们在这里关注更具挑战性的硬约束情况，即部分满意的客户不会获得利润。例如，当客户端出于容错的原因想要将文件存储在 $$\kappa (c)$$κ(c) 或更多附近的服务器时，此模型将进行建模。文献中的其他动机包括 4G 蜂窝网络中的资源分配和具有分配限制的相关机器上的机器调度。在 $$k$$k-SA 的 r 限制版本中，没有客户端需要超过任何相邻服务器容量的 r 部分。我们为 r-restricted $$k$$k-SA 提出了一个（集中式）多项式时间近似算法。该算法的一个变体在给定资源增加因子 $$1+r$$1+r 时实现了近似比率。我们使用后一个结果来提出 $$k$$k-SA 的近似算法。在分布式设置中，我们提出：（i）r-restricted $$k$$k-SA 的近似算法，（ii）使用资源增强因子 $$1+r$$1+r 的近似算法对于 r-restricted $$k$$k-SA，对于任何常数 $$\varepsilon >0$$ε>0，(iii) $$k$$k-SA 的近似算法（预期）。这三种分布式算法在 $$O(k^2 \varepsilon ^{-2} \log ^3 n)$$O(k2ε-2log3n) 同步轮次中运行（概率很高）。特别是，这产生了 $$1$$1-SA 的第一个分布式近似值。