Combinatorial Philosopher Inequalities
Enze Sun, Zhihao Gavin Tang, Yifan Wang
TL;DR
The paper investigates online combinatorial allocation with stochastic agents, aiming to beat the classical prophet-inequality benchmark by designing approximations against the online optimum. It introduces an online configuration LP relaxation that upper-bounds the online optimum and develops a submodular-focused philosopher inequality achieving a $0.5+Ω(1)$-approximation, together with a novel half-double sampling rounding that attains at least $0.5625$ of the LP on favorable instances. A key conceptual advance is the free-deterministic decomposition, which partitions the LP’s value into a near-free part and a deterministic part, guiding both the easy-instance reduction and the specialized rounding. Conversely, the work proves a structural barrier for XOS valuations by showing a $0.5$ integrality gap for the online LP, highlighting limitations of current LP relaxations and signaling the need for new relaxations to extend philosopher-inequality results beyond submodular valuations. Overall, the results advance the understanding of online configurational relaxations, provide poly-time algorithms with improved approximation factors, and delineate the gap between submodular and XOS valuations in the online setting with prophet-like benchmarks.
Abstract
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 $0.5 + Ω(1)$ 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.