Inspired by equity considerations, we consider top-$k$ norm, ordered norm, and symmetric monotonic norm objectives for various combinatorial optimization problems. Top-$k$ norms and ordered norms have natural interpretations in terms of minimizing the impact on individuals bearing largest costs. To model decision-making with multiple equity criteria, we study the notion of portfolios of solutions with the property that each norm or equity criteria has an approximately optimal solution in this portfolio. We attempt to characterize portfolios by their sizes and approximation factor guarantees for various combinatorial problems. For a given problem, we investigate whether (1) there exists a single solution that is approximately optimal for all norms, (2) there exists a small approximately optimal portfolio of size larger than 1, (3) there exist polynomial time algorithms to find these small portfolios. We study an algorithmic framework to obtain single solutions that are approximately optimal for all norms. We show the existence of such a solution for problems such as $k$-clustering, ordered set cover, scheduling for job completion time minimization, and scheduling for machine load minimization on identical machines. We also give efficient algorithms to find these solutions in most cases, except set cover where we show there is a gap in terms of computational complexity. Our work improves upon the best-known approximation factor across all norms for a single solution in $k$-clustering. For uncapacitated facility location and scheduling for machine load minimization with identical jobs, we obtain logarithmic sized portfolios, also providing a matching lower bound in the latter case. Our work results in new open combinatorial questions, which might be of independent interest.

翻译：受公平性考量启发，我们针对多种组合优化问题研究了top-$k$范数、有序范数及对称单调范数目标。top-$k$范数和有序范数在最小化承担最大成本个体的影响方面具有自然解释。为建模多公平性准则下的决策，我们研究了具有如下性质的解集组合：该组合中每个范数或公平性准则均存在近似最优解。我们尝试通过组合的规模及其对多种组合问题的近似比保证来刻画该组合。针对给定问题，我们探究：(1)是否存在对所有范数均近似最优的单一解；(2)是否存在规模大于1的较小近似最优组合；(3)是否存在多项式时间算法寻找这些较小组合。我们研究了一种获取对所有范数均近似最优的单一解的算法框架，证明此类解存在于$k$-聚类、有序集合覆盖、最小化工件完工时间的调度以及同构机器最小化机器负载的调度等问题中。除集合覆盖问题外，我们在大多数情况下给出了求解这些解的高效算法——对于集合覆盖问题，我们证明其计算复杂度存在间隙。我们的工作改进了$k$-聚类中单一解在所有范数下的最佳已知近似比。针对无容量限制设施选址问题及同构工件的最小化机器负载调度问题，我们获得了对数规模组合，并在后者中给出了匹配的下界。本文工作提出了可能具有独立研究价值的新型开放组合学问题。