Modern Primal-Dual Frameworks for Prior-Free Online Resource Allocation

2026-06-13

AI Summary

This work addresses adversarial online resource allocation problems without prior information, where conventional linear programming (LP) relaxations often fail under stochastic settings. The paper introduces two dual analysis frameworks: the first leverages regularized convex optimization and KKT conditions within an LP-based approach, while the second establishes a general dual certificate mechanism that does not rely on linear programming. This latter framework unifies the treatment of complex scenarios—including reusable resources, stochastic rewards, and whole-page optimization—and provides a generic template for proving competitive ratios. By offering a unified theoretical foundation for a broad class of online matching and resource allocation problems, the proposed framework not only validates existing algorithms but also facilitates the design of new ones, delivering strong theoretical guarantees across diverse models.

Abstract

Linear-programming (LP)-based primal-dual methods are fundamental for designing and analyzing algorithms in adversarial (prior-free) online resource allocation. This chapter provides a tutorial on two modern primal-dual frameworks, emphasizing recent developments and contemporary models in operations research. Part~I develops an LP-based convex-programming framework where solving a regularized convex program at each arrival captures the tradeoff between greediness and hedging, yielding a dual certificate via Karush-Kuhn-Tucker (KKT) conditions. Because standard LP relaxations can be weak or intractable for stochastic outcomes, Part~II introduces a complementary LP-free framework that provides a universal certificate system for evaluating competitive ratios under such uncertainty. Covering a wide array of models -- including online vertex-weighted bipartite matching, edge-weighted online matching with free disposal, online matching with stochastic rewards, reusable resources, two-sided assortment optimization, configuration allocation (whole-page optimization), AdWords, and costly cancellations -- the tutorial equips readers with versatile proof templates to analyze existing algorithms and develop new solutions for emerging applications.

Problem

• online resource allocation
• primal-dual methods
• competitive ratio
• stochastic rewards
• adversarial setting

Innovation

• primal-dual framework
• online resource allocation
• convex programming
• competitive ratio
• LP-free certificate