负载均衡问题的可加近似格式
Additive Approximation Schemes for Load Balancing Problems
摘要：
在本文中，我们引入了加法近似方案的概念，并将其应用于负载均衡问题。加法近似方案的目的是找到一个解决方案，在最多$εh$的目标，为一些合适的参数$h$的绝对误差。在参数$h$提供下限的情况下，加性近似方案意味着标准的乘性近似方案，并且当$h \ll$ OPT.On另一方面，当不存在PTAS（或不太可能存在）时，加性近似方案可以提供不同的近似概念。我们考虑的问题，分配工作的机器与上下界的机器的负载相同的机器。这种设置概括了最大完工时间最小化、圣诞老人问题（在相同的机器上）和最小化嫉妒的圣诞老人问题。对于最后一个问题，其目标是最小化最大和最小负载之间的差，最优目标值可能为零，因此获得任何乘法近似保证是NP难的。对于这类问题，我们提出了添加剂近似计划的$h = p_{\max}$，最大加工时间的工作。我们的技术贡献是双重的。首先，我们引入了一个新的放松的基础上，整体分配插槽的机器和分数分配工作的插槽（插槽MILP）。我们确定的插槽MILP，这使我们能够有效地解决它的（近）最优解的结构特性，假设有$O(1)$不同的下限和上限的机器负载（这是上述三个问题的相关设置）。第二个技术贡献是基于局部搜索的算法，其将解舍入到时隙MILP，在至多$ε\cdot p_{\max}$的目标负载间隔上引入附加误差。
英文摘要：
In this paper we introduce the concept of additive approximation schemes and apply it to load balancing problems. Additive approximation schemes aim to find a solution with an absolute error in the objective of at most $εh$ for some suitable parameter $h$. In the case that the parameter $h$ provides a lower bound an additive approximation scheme implies a standard multiplicative approximation scheme and can be much stronger when $h \ll$ OPT. On the other hand, when no PTAS exists (or is unlikely to exist), additive approximation schemes can provide a different notion for approximation. We consider the problem of assigning jobs to identical machines with lower and upper bounds for the loads of the machines. This setting generalizes problems like makespan minimization, the Santa Claus problem (on identical machines), and the envy-minimizing Santa Claus problem. For the last problem, in which the objective is to minimize the difference between the maximum and minimum load, the optimal objective value may be zero and hence it is NP-hard to obtain any multiplicative approximation guarantee. For this class of problems we present additive approximation schemes for $h = p_{\max}$, the maximum processing time of the jobs. Our technical contribution is two-fold. First, we introduce a new relaxation based on integrally assigning slots to machines and fractionally assigning jobs to the slots (the slot-MILP). We identify structural properties of (near-)optimal solutions of the slot-MILP, which allow us to solve it efficiently, assuming that there are $O(1)$ different lower and upper bounds on the machine loads (which is the relevant setting for the three problems mentioned above). The second technical contribution is a local-search based algorithm which rounds a solution to the slot-MILP introducing an additive error on the target load intervals of at most $ε\cdot p_{\max}$.
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