Motivated by the classic Generalized Assignment Problem, we consider the Graph Balancing problem in the presence of orientation costs: given an undirected multi-graph G = (V,E) equipped with edge weights and orientation costs on the edges, the goal is to find an orientation of the edges that minimizes both the maximum weight of edges oriented toward any vertex (makespan) and total orientation cost. We present a general framework for minimizing makespan in the presence of costs that allows us to: (1) achieve bicriteria approximations for the Graph Balancing problem that capture known previous results (Shmoys-Tardos [Math. Progrm. 93], Ebenlendr-Krc\'al- Sgall [Algorithmica 14], and Wang-Sitters [Inf. Process. Lett. 16]); and (2) achieve bicriteria approximations for extensions of the Graph Balancing problem that admit hyperedges and unrelated weights. Our framework is based on a remarkably simple rounding of a strengthened linear relaxation. We complement the above by presenting bicriteria lower bounds with respect to the linear programming relaxations we use that show that a loss in the total orientation cost is required if one aims for an approximation better than 2 in the makespan.

中文翻译：

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图平衡与定向成本

受经典广义分配问题的启发，我们考虑了存在定向成本的图平衡问题：给定一个无向多图 G = (V,E) 配备边权重和边上的定向成本，目标是找到边的方向，使朝向任何顶点的边的最大权重（makespan）和总方向成本最小化。我们提出了一个在存在成本的情况下最小化完工时间的通用框架，它使我们能够：（1）实现图平衡问题的双标准近似，捕获已知的先前结果（Shmoys-Tardos [Math. Progrm. 93], Ebenlendr-Krc\ 'al-Sgall [Algorithmica 14] 和 Wang-Sitters [Inf. Process. Lett. 16])；(2) 实现图平衡问题扩展的双准则近似，允许超边和不相关的权重。我们的框架基于一个非常简单的强化线性松弛四舍五入。我们通过提出关于我们使用的线性规划松弛的双准则下界来补充上述内容，这表明如果一个人的目标是在 makespan 中的近似值优于 2，则需要总定向成本的损失。