The Online Submodular Assignment Problem
Daniel Hathcock, Billy Jin, Kalen Patton, Sherry Sarkar, Michael Zlatin
TL;DR
This paper introduces Online Submodular Assignment Problem (Online SAP), a comprehensive framework unifying online matching, AdWords, GAP, laminar allocations, matroid coloring, and related online optimization problems under submodular constraints. It develops a water-level machinery that generalizes the water-filling paradigm to polymatroids, enabling a continuous pricing-based allocation and a robust primal–dual analysis. The authors prove a deterministic $(1-1/e)$-competitive fractional algorithm for Online SAP, with a further integral algorithm under a small-bids assumption achieving $(1-1/e- ext{ε})$-competitiveness, and they extend the framework to Online Submodular Welfare Maximization for matroid rank valuations, achieving $(1-1/e)$ in expectation. The key contributions include three equivalent formulations of water levels, a deep connection to submodular utility markets (SUA) and principal partitions, and broad applicability to a wide class of online allocation problems, offering a versatile toolkit for designing optimal online algorithms under submodular constraints.
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.