The Online Submodular Assignment Problem

TL;DR

This paper introduces the submodular assignment problem (SAP), a broad generalization of online resource allocation problems, and provides competitive algorithms for both fractional and integral cases, leveraging a novel water level vector construction.

Contribution

It presents a unified framework for online assignment problems, introduces a competitive fractional algorithm, and develops new structural tools for integral solutions.

Findings

Fractional algorithm is (1-1/e)-competitive.

Integral algorithm under small-bids assumption is (1-1/e-epsilon)-competitive.

New water level vector construction generalizes water-filling paradigm.

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 (1-1/e)-competitive. Additionally, we study several integral special cases of the problem. In particular, we provide a (1-1/e-epsilon)-competitive integral algorithm under a small-bids assumption, and a (1-1/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.