Online Submodular Welfare Maximization with Post-Allocation Stochasticity and Reusability

Core Concepts

Non-adaptive Greedy algorithm achieves competitive ratio against adaptive offline benchmark in various arrival models.

Abstract

Introduction:

Problem of online submodular welfare maximization (OSW).
Adversarial and unknown IID stochastic arrival models.

Online Submodular Welfare Maximization:

Greedy algorithm's competitive ratio.
Adaptivity to stochastic rewards.

Generalizations:

Settings beyond OSW.
Reusable resources and post-allocation stochasticity.

Contributions:

Online submodular order welfare maximization (OSOW).
Greedy algorithm's competitiveness.

Preliminaries and New Models:

Definitions and key concepts.

Main Results:

Non-adaptive Greedy's competitiveness.

Theoretical Analysis:

Theoretical insights and implications.

Stats

Greedy algorithm achieves the highest possible competitive ratio of 0.5.
Unless NP=RP, no polynomial time online algorithm for OSW has a better competitive ratio.
Greedy algorithm is 0.5 competitive for OSW.

Quotes

"In general, adaptivity to stochastic rewards offers no theoretical benefits."

Key Insights Distilled From

Online Submodular Welfare Maximization Meets Post-Allocation Stochasticity and Reusability

by Rajan Udwani at arxiv.org 03-28-2024

https://arxiv.org/pdf/2403.18059.pdf

Deeper Inquiries

Is there a model that captures new features such as reusability and post-allocation stochasticity while maintaining the richness of OSW

The model that captures new features such as reusability and post-allocation stochasticity while maintaining the richness of Online Submodular Welfare Maximization (OSW) is the Online Submodular Order Welfare Maximization (OSOW). In this model, the objective functions are monotone submodular order functions, which allow for the incorporation of reusability and post-allocation stochasticity. By defining submodular order functions that capture the nonlinear and combinatorial aspects of the problem, OSOW extends the capabilities of OSW to handle these new features effectively.

Does non-adaptive Greedy maintain its competitive ratio in a general setting with post-allocation stochasticity

Non-adaptive Greedy maintains its competitive ratio in a general setting with post-allocation stochasticity. In the context provided, it has been shown that non-adaptive Greedy is 0.5 competitive against an adaptive offline benchmark in the presence of post-allocation stochasticity. This means that even in scenarios where the outcome of a proposed match is uncertain or delayed, non-adaptive Greedy can achieve a competitive ratio of 0.5, demonstrating its effectiveness in handling stochastic rewards and similar features.

How can combinatorial actions be effectively captured in a unified manner in online submodular welfare maximization

Combinatorial actions can be effectively captured in a unified manner in online submodular welfare maximization by considering a broader class of functions known as submodular order functions. These functions extend the concept of submodularity to capture the interactions and dependencies between multiple resources and arrivals in a combinatorial setting. By defining objective functions that adhere to the submodular order property, the model can accommodate various forms of combinatorial actions, providing a comprehensive framework for optimizing resource allocation in complex scenarios.