Randomized Rounding Approaches to Online Allocation, Sequencing, and Matching

TL;DR

This paper introduces randomized rounding techniques for designing online policies in stochastic optimization problems, addressing challenges of sequential decision-making under uncertainty.

Contribution

It provides a tutorial with four examples demonstrating how randomized rounding can be applied to online allocation, sequencing, and matching problems.

Findings

Effective online policies are constructed using randomized rounding.

The approach preserves prescribed probabilities in expectation despite constraints.

Applications include contention resolution, probing, knapsack, and matching problems.

Abstract

Randomized rounding is a technique that was originally used to approximate hard offline discrete optimization problems from a mathematical programming relaxation. Since then it has also been used to approximately solve sequential stochastic optimization problems, overcoming the curse of dimensionality. To elaborate, one first writes a tractable linear programming relaxation that prescribes probabilities with which actions should be taken. Rounding then designs a (randomized) online policy that approximately preserves all of these probabilities, with the challenge being that the online policy faces hard constraints, whereas the prescribed probabilities only have to satisfy these constraints in expectation. Moreover, unlike classical randomized rounding for offline problems, the online policy's actions unfold sequentially over time, interspersed by uncontrollable stochastic realizations that affect the feasibility of future actions. This tutorial provides an introduction for using randomized rounding to design online policies, through four self-contained examples representing fundamental problems in the area: online contention resolution, stochastic probing, stochastic knapsack, and stochastic matching.