Constrained maximization of submodular functions poses a central problem in combinatorial optimization. In many realistic scenarios, a number of agents need to maximize multiple submodular objectives over the same ground set. We study such a setting, where the different solutions must be disjoint, and thus, questions of fairness arise. Inspired from the fair division literature, we suggest a simple round-robin protocol, where agents are allowed to build their solutions one item at a time by taking turns. Unlike what is typical in fair division, however, the prime goal here is to provide a fair algorithmic environment; each agent is allowed to use any algorithm for constructing their respective solutions. We show that just by following simple greedy policies, agents have solid guarantees for both monotone and non-monotone objectives, and for combinatorial constraints as general as $p$-systems (which capture cardinality and matroid intersection constraints). In the monotone case, our results include approximate EF1-type guarantees and their implications in fair division may be of independent interest. Further, although following a greedy policy may not be optimal in general, we show that consistently performing better than that is computationally hard.

翻译：子模函数的约束最大化是组合优化中的核心问题。在许多现实场景中，多个主体需要在同一基集上最大化多个子模目标函数。我们研究这类问题，其中不同解必须互不相交，因此产生了公平性问题。受公平分配文献启发，我们提出一种简单的轮询协议，允许主体轮流逐项构建其解集。然而，与典型公平分配不同，本研究的主要目标是提供公平的算法环境：每个主体可使用任意算法构建其相应解集。我们证明，仅通过遵循简单贪婪策略，主体即可在单调和非单调目标函数以及一般性组合约束（如$p$-系统，包括基数约束和拟阵交约束）下获得可靠保障。在单调情形下，我们的结果包含近似EF1型保障，其公平分配含义可能具有独立研究价值。此外，尽管遵循贪婪策略通常并非最优，但我们证明持续超越该策略在计算上是困难的。