[论文解读] On the Configuration-LP for Scheduling on Unrelated Machines

José Verschae, Andreas Wiese|arXiv (Cornell University)|Jan 1, 2010

Scheduling and Optimization Algorithms参考文献 18被引用 2

一句话总结

本文研究了无关机器上配置-LP的整数性间隙，证明了其在无关图平衡问题中的整数性间隙为2，突显了其与受限分配情形（该情形下间隙至多为33/17）之间的关键差异。此外，本文提出了一种组合式的2-近似算法用于MaxMin平衡问题，其近似因子与目前已知最佳结果一致，并优于以往基于LP的方法。

ABSTRACT

One of the most important open problems in machine scheduling is the problem of scheduling a set of jobs on unrelated machines to minimize the makespan. The best known approximation algorithm for this problem guarantees an approximation factor of 2. It is known to be NP-hard to approximate with a better ratio than 3/2. Closing this gap has been open for over 20 years. The best known approximation factors are achieved by LP-based algorithms. The strongest known linear program formulation for the problem is the configuration-LP. We show that the configuration-LP has an integrality gap of 2 even for the special case of unrelated graph balancing, where each job can be assigned to at most two machines. In particular, our result implies that a large family of cuts does not help to diminish the integrality gap of the canonical assignment-LP. Also, we present cases of the problem which can be approximated with a better factor than 2. They constitute valuable insights for constructing an NP-hardness reduction which improves the known lowerbound. Very recently Svensson [22] studied the restricted assignment case, where each job can only be assigned to a given set of machines on which it has the same processing time. He shows that in this setting the configuration-LP has an integrality gap of 33/17≈1.94. Hence, our result imply that the unrelated graph balancing case is significantly more complex than the restricted assignment case. Then we turn to another objective function: maximizing the minimum machine load. For the case that every job can be assigned to at most two machines we give a purely combinatorial 2-approximation which is best possible, unless P=NP. This improves on the computationally costly LP-based (2 +ε)-approximation algorithm by Chakrabarty et al. [7].

研究动机与目标

• 确定无关图平衡问题中配置-LP的整数性间隙。
• 比较无关图平衡问题与受限分配情形的复杂性。
• 为MaxMin平衡问题设计更优的近似算法。
• 探索可实现更优近似因子的可 tractable 特例。

提出的方法

• 通过构造一类实例，证明了配置-LP在无关图平衡问题中的整数性间隙为2，其中LP解仅为最优整数解的一半。
• 提出一种基于预分配和由可分配至两台机器的作业导出的二分图的顶点着色的组合式2-近似算法，用于MaxMin平衡问题。
• 采用预分配阶段，固定必须分配至特定机器以满足目标负载T的作业。
• 构建一个图，其中顶点表示分配至某台机器的作业，边连接可分配至两台机器的作业或同一台机器上连续的作业，从而形成一个二分图。
• 对图进行2-着色，以确保每台机器在其活跃集合中获得剩余作业总权重的一半以上。
• 将LST舍入技术适配用于生成MaxMin分配的半整数解，确保负载损失被控制在常数因子内。

实验结果

研究问题

• RQ1 无关图平衡问题中配置-LP的整数性间隙是多少？
• RQ2 无关图平衡问题的整数性间隙与受限分配问题相比如何？
• RQ3 能否设计一种完全组合式的2-近似算法用于MaxMin平衡问题，从而避免昂贵的LP求解？
• RQ4 在哪些MaxMin分配的可 tractable 情况下，可以实现更优的近似因子？
• RQ5 能否高效计算半整数解，以在MaxMin分配中保持常数因子近似？

主要发现

• 配置-LP在无关图平衡问题中的整数性间隙为2，尽管在受限分配情形中该间隙至多为33/17。
• 由于整数性间隙更高，无关图平衡问题比受限分配情形复杂得多。
• 本文提出了一种完全组合式的2-近似算法用于MaxMin平衡问题，该算法在P ≠ NP的前提下为最优。
• 该算法时间复杂度为O(|I|²)，避免了基于LP方法的计算开销。
• 半整数解可在多项式时间内计算，使得目标值至少为最优整数解的一半。
• 对于大作业数量为常数的实例，或除O(log n)台机器外其余机器的大作业分配已知的实例，可实现2-近似。