In online combinatorial allocation, agents arrive sequentially and items are allocated in an online manner. The algorithm designer only knows the distribution of each agent's valuation, while the actual realization of the valuation is revealed only upon her arrival. Against the offline benchmark, Feldman, Gravin, and Lucier (SODA 2015) designed an optimal $0.5$-competitive algorithm for XOS agents. An emerging line of work focuses on designing approximation algorithms against the (computationally unbounded) optimal online algorithm. The primary goal is to design algorithms with approximation ratios strictly greater than $0.5$, surpassing the impossibility result against the offline optimum. Positive results are established for unit-demand agents (Papadimitriou, Pollner, Saberi, Wajc, MOR 2024), and for $k$-demand agents (Braun, Kesselheim, Pollner, Saberi, EC 2024). In this paper, we extend the existing positive results for agents with submodular valuations by establishing a $0.5 + \Omega(1)$ approximation against a newly constructed online configuration LP relaxation for the combinatorial allocation setting. Meanwhile, we provide negative results for agents with XOS valuations by providing a $0.5$ integrality gap for the online configuration LP, showing an obstacle of existing approaches.

翻译：在线组合分配问题中，智能体按序到达，物品以在线方式分配。算法设计者仅知晓各智能体估值的概率分布，而估值实际实现值仅在其到达时揭示。针对离线基准，Feldman、Gravin与Lucier（SODA 2015）为XOS智能体设计了最优的$0.5$竞争比算法。近期一系列研究聚焦于设计针对（计算无界的）最优在线算法的近似算法，其核心目标是设计近似比严格大于$0.5$的算法，以突破针对离线最优解的不可能性结果。当前已在单位需求智能体（Papadimitriou、Pollner、Saberi、Wajc，MOR 2024）与$k$需求智能体（Braun、Kesselheim、Pollner、Saberi，EC 2024）场景中取得积极成果。本文通过针对组合分配场景中新构建的在线配置线性规划松弛，建立$0.5 + \Omega(1)$的近似比，将现有积极结果扩展至具有次模估值函数的智能体。同时，我们通过构建在线配置线性规划的$0.5$整数间隙，为XOS估值智能体提供负面结果，揭示了现有方法面临的障碍。