[论文解读] Robust Assignments with Vulnerable Nodes

David Adjiashvili, Viktor Bindewald+1 more|arXiv (Cornell University)|Mar 17, 2017

Complexity and Algorithms in Graphs参考文献 4被引用 3

一句话总结

本文建立了集合覆盖（SC）问题与易损鲁棒分配问题（V-RAP）之间的形式化对应关系，表明V-RAP中的可行解与SC中的可行集合覆盖一一对应，且保持可行性与成本一致。其核心贡献在于提出一种归约方法，使得可通过V-RAP求解SC问题，同时保持最优解结构与成本等价性。

ABSTRACT

Various real-life planning problems require making upfront decisions before all parameters of the problem have been disclosed. An important special case of such problem especially arises in scheduling and staff rostering problems, where a set of tasks needs to be assigned to an available set of resources (personnel or machines), in a way that each task is assigned to one resource, while no task is allowed to share a resource with another task. In its nominal form, the resulting computational problem reduces to the well-known assignment problem that can be modeled as matching problems on bipartite graphs. In recent work \cite{adjiashvili_bindewald_michaels_icalp2016}, a new robust model for the assignment problem was introduced that can deal with situations in which certain resources, i.e.\ nodes or edges of the underlying bipartite graph, are vulnerable and may become unavailable after a solution has been chosen. In the original version from \cite{adjiashvili_bindewald_michaels_icalp2016} the resources subject to uncertainty are the edges of the underlying bipartite graph. In this follow-up work, we complement our previous study by considering nodes as being vulnerable, instead of edges. The goal is now to choose a minimum-cost collection of nodes such that, if any vulnerable node becomes unavailable, the remaining part of the solution still contains sufficient nodes to perform all tasks. From a practical point of view, such type of unavailability is interesting as it is typically caused e.g.\ by an employee's sickness, or machine failure. We present algorithms and hardness of approximation results for several variants of the problem.

研究动机与目标

• 形式化集合覆盖（SC）问题与易损鲁棒分配问题（V-RAP）之间的关系。
• 证明V-RAP中的可行解与SC中的可行集合覆盖完全对应。
• 在从SC到V-RAP的转换过程中保持解的成本。
• 为通过V-RAP将鲁棒优化方法应用于SC提供理论基础。
• 确立V-RAP中的不确定集对应于SC中待覆盖的元素。

提出的方法

• 通过转换步骤（T1）–（T2），从给定的SC实例（[k], S）构造一个V-RAP实例（G, F, c）。
• 将图G定义为两个不相交集合的并集：U[k] ∪ (WS ∪ W[k])，以及边集E[k] ∪ E_SC。
• 设定不确定集F = W[k]，表示V-RAP公式中的易损节点。
• 根据公式（LABEL:eq_redcostfunc）定义成本向量c ∈ R^{WS ∪ W[k]} ≥ 0。
• 在满足W[k] ⊆ X的可行集合X ⊆ WS ∪ W[k]与SC中的可行集合覆盖CX ⊆ S之间建立双射关系。
• 证明V-RAP中X的成本等于SC中CX的成本，确保成本保持不变。

实验结果

研究问题

• RQ1 如何在保持可行性与成本的前提下，将集合覆盖问题归约为易损鲁棒分配问题？
• RQ2 哪些结构性质确保V-RAP中的每个可行解都对应于SC中的一个可行集合覆盖？
• RQ3 在何种条件下，从SC到V-RAP的转换能保持成本等价性？
• RQ4 节点集U[k]、WS与W[k]在V-RAP框架中如何共同建模SC问题？
• RQ5 不确定集F = W[k]在确保归约正确性方面发挥什么作用？

主要发现

• 对于任意满足W[k] ⊆ X的V-RAP实例中的可行解X，其在原始SC实例中均存在对应的可行集合覆盖CX ⊆ S。
• 当且仅当ws在V-RAP实例中与wSj ∈ X匹配时，SC中的每个元素s ∈ [k] 被某个Sj ∈ CX 覆盖。
• V-RAP实例中X的成本与SC实例中CX的成本完全相同，确保了成本保持不变。
• 对于满足W[k] ⊆ X的可行解，从SC到V-RAP的转换是双射的。
• 证明确认了通过辅助图中的匹配条件，V-RAP公式能正确建模SC问题。
• 在G、F与c的指定构造下，该归约成立，为后续算法应用建立了可靠的理论基础。