The Online Submodular Assignment Problem

2026-04-11

AI Summary

This paper studies the online submodular allocation problem (SAP), unifying classical settings such as bipartite matching, flow scheduling, and layered allocation. We introduce a novel polyhedral construction based on “water-level vectors,” generalizing the classic water-filling paradigm to matroids and general convex polytopes, and uncovering structural connections between submodular utility allocation and principal partition sequences. Our main contributions are threefold: (i) We design the first online algorithm achieving the optimal fractional competitive ratio of $1-1/e$; (ii) Under the small-bid assumption, we obtain an integer competitive ratio of $1-1/e-varepsilon$ via perturbation analysis and polynomial-time approximation; (iii) For submodular welfare maximization modeled by matroid rank functions, we achieve the exact integer competitive ratio $1-1/e$. The results advance the theoretical understanding of online submodular optimization and extend water-filling techniques beyond separable concave objectives to combinatorial and geometric domains.

Technology Category

Machine Learning: Online Learning & BanditsSearch and Optimization: Mixed Discrete/Continuous SearchGame Theory and Economic Paradigms: Mechanism Design

Application Category

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

Abstract

Online resource allocation is a rich and varied field. One of the most well-known problems in this area is online bipartite matching, introduced in 1990 by Karp, Vazirani, and Vazirani [KVV90]. Since then, many variants have been studied, including AdWords, the generalized assignment problem (GAP), and online submodular welfare maximization. In this paper, we introduce a generalization of GAP which we call the submodular assignment problem (SAP). This generalization captures many online assignment problems, including all classical online bipartite matching problems as well as broader online combinatorial optimization problems such as online arboricity, flow scheduling, and laminar restricted allocations. We present a fractional algorithm for online SAP that is $(1-frac{1}{e})$-competitive. Additionally, we study several integral special cases of the problem. In particular, we provide a $(1-frac{1}{e}-ε)$-competitive integral algorithm under a small-bids assumption, and a $(1-frac{1}{e})$-competitive integral algorithm for online submodular welfare maximization where the utility functions are given by rank functions of matroids. The key new ingredient for our results is the construction and structural analysis of a "water level" vector for polymatroids, which allows us to generalize the classic water-filling paradigm used in online matching problems. This construction reveals connections to submodular utility allocation markets and principal partition sequences of matroids.

Problem

• Online Resource Allocation
• Submodular Assignment Problem (SAP)
• Efficiency Maximization

Innovation

• Online Submodular Assignment Problem
• Water-Filling Technique
• Welfare Allocation