Combinatorial Philosopher Inequalities

2025-10-11

AI Summary

This paper studies approximation algorithms for online combinatorial allocation with submodular and XOS valuation agents. For submodular valuations, we propose a novel framework based on the online configuration linear programming (LP) relaxation—the first of its kind—combined with stochastic analysis and a constructive proof of an integrality gap strictly greater than 0.5, achieving a competitive ratio of $0.5 + Omega(1)$ and thus breaking the long-standing $0.5$ barrier; this ratio further surpasses known offline approximation limits. For XOS valuations, we establish a fundamental limitation of the same framework: we rigorously prove that the online configuration LP exhibits an integrality gap of exactly $0.5$, thereby demonstrating an inherent barrier to achieving any competitive ratio exceeding $0.5$ in the XOS setting. Collectively, our work advances the theoretical frontier of online submodular optimization and precisely characterizes the approximability boundary for XOS valuations.

Technology Category

• Search and Optimization: Algorithm Configuration
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Online Learning & Bandits

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Extends approximation algorithms for submodular valuations in online allocation
• Establishes positive results against online configuration LP relaxation
• Provides negative results showing limitations for XOS valuations

Innovation

• Online configuration LP relaxation for submodular valuations
• Achieving approximation ratio strictly above 0.5
• Establishing integrality gap for XOS valuations