The Online Submodular Assignment Problem

Online resource allocation is a rich and var-ied field. One of the most well-known problems in this area is online bipartite matching, introduced in 1990 by Karp, Vazirani, and Vazirani. 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 $(1\ -1/e-\varepsilon){-}$ 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.

Authors

• Daniel Hathcock
• Billy Jin
• Kalen Patton
• Sherry Sarkar
• Michael Zlatin

Keywords

• Computer science
• Vectors
• Partitioning algorithms
• Resource management
• Optimization
• Assignment Problem
• Water Level
• Well-known Problem
• Competitive Algorithm
• Bipartite Matching
• Welfare Maximization
• Infinitesimal
• Objective Value
• Time Slot
• Algorithm For Problem
• Marginal Utility
• Change Objectives
• Dual Variables
• Algorithm Running
• Ranking Algorithm
• Random Bits
• Dual Objective
• Dual Solution
• Feasibility Indicators
• Small Assumption
• Competitive Ratio
• Dual Value
• Combinatorial Optimization
• Online Algorithms
• Adwords
• Submodular Functions