Constrained maximization of submodular functions is a central problem in combinatorial optimization. In many realistic scenarios, multiple agents each need to maximize their own submodular objective over a common ground set, subject to individual constraints, with the requirement that their solutions be disjoint. We study this setting through the lens of algorithmic fairness and constrained fair division. Inspired by the fair division literature, we propose and analyze a simple Round-Robin protocol in which agents take turns building their solutions one item at a time; each agent is free to use any internal algorithm, and the protocol itself performs no computation. We show that agents following simple greedy policies enjoy solid guarantees for both monotone and non-monotone objectives subject to constraints as general as $p$-systems. For monotone objectives, a greedy agent $i$ with a $p_i$-system constraint achieves a $1/(n+p_i)$ fraction of the best value available when they first get to choose. On instances that are robust to competition -- where no agent's optimal value is greatly affected by losing some items to others -- these guarantees improve to a $1/Θ(p_i)$ approximation of the unconstrained optimum, which is asymptotically best-possible in polynomial time. We further establish novel fairness guarantees: greedy agents produce approximately feasible-envy-free-up-to-one-item (FEF1) and approximately feasible-envy-free-towards-unallocated-items (FEFu) allocations for monotone and non-monotone objectives. Via a simple augmented protocol and a self-contained polynomial-time proxy algorithm, we also obtain the first $Θ(1/p_i)$-approximate feasible maximin share (FMMS) guarantees for submodular agents with combinatorial constraints. Finally, although greedy policies may not be individually optimal, consistently improving upon them is NP-hard even in the simplest settings.

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，多个智能体各自需要在公共基础集上最大化其子模目标，同时受限于个体约束，并要求其解互不相交。我们通过算法公平性和约束公平划分的视角研究该设置。受公平划分文献的启发，我们提出并分析了一种简单的轮询协议：智能体轮流逐个构建其解集；每个智能体可自由使用任意内部算法，而协议本身不执行任何计算。我们证明，遵循简单贪婪策略的智能体在单调和非单调目标下，对于一般性约束（如$p$-系统）均具有可靠保证。对于单调目标，受$p_i$-系统约束的贪婪智能体$i$在首次获得选择权时，可获得其最优值$1/(n+p_i)$的近似。在鲁棒性竞争实例中（即无智能体的最优值因其他智能体获得部分物品而大幅下降），这些保证可提升至无约束最优的$1/Θ(p_i)$近似，这在多项式时间内渐近最优。我们进一步建立了新的公平性保证：对于单调和非单调目标，贪婪智能体产生的分配近似满足可行无嫉妒至多一件物品（FEF1）和可行无嫉妒至未分配物品（FEFu）。通过简单增强协议和自包含的多项式时间代理算法，我们首次为受组合约束的子模智能体提供了$Θ(1/p_i)$-近似可行最大最小份额（FMMS）保证。最后，尽管贪婪策略可能并非个体最优，但即使设定最简单，持续改进贪婪策略也是NP-hard的。