# Near Optimal Online Algorithms and Fast Approximation Algorithms for Resource Allocation Problems

## TL;DR

This paper develops near-optimal online algorithms for resource allocation that adapt to unknown distributions and adversarial inputs, providing fast approximation solutions for large-scale LPs with broad applications.

## Contribution

It introduces a unified framework for robust online resource allocation algorithms that work under both stochastic and adversarial models, with near-optimal tradeoffs and fast LP-solving methods.

## Key findings

- Achieved a $1- ext{epsilon}$ competitive ratio for online algorithms under unknown distributions.
- Designed algorithms that perform well even with adversarially controlled request distributions.
- Provided fast approximation algorithms for large LPs with packing and covering constraints.

## Abstract

We present prior robust algorithms for a large class of resource allocation problems where requests arrive one-by-one (online), drawn independently from an unknown distribution at every step. We design a single algorithm that, for every possible underlying distribution, obtains a $1-\epsilon$ fraction of the profit obtained by an algorithm that knows the entire request sequence ahead of time. The factor $\epsilon$ approaches $0$ when no single request consumes/contributes a significant fraction of the global consumption/contribution by all requests together. We show that the tradeoff we obtain here that determines how fast $\epsilon$ approaches $0$, is near optimal: we give a nearly matching lower bound showing that the tradeoff cannot be improved much beyond what we obtain. Going beyond the model of a static underlying distribution, we introduce the adversarial stochastic input model, where an adversary, possibly in an adaptive manner, controls the distributions from which the requests are drawn at each step. Placing no restriction on the adversary, we design an algorithm that obtains a $1-\epsilon$ fraction of the optimal profit obtainable w.r.t. the worst distribution in the adversarial sequence. In the offline setting we give a fast algorithm to solve very large LPs with both packing and covering constraints. We give algorithms to approximately solve (within a factor of $1+\epsilon$) the mixed packing-covering problem with $O(\frac{\gamma m \log (n/\delta)}{\epsilon^2})$ oracle calls where the constraint matrix of this LP has dimension $n\times m$, the success probability of the algorithm is $1-\delta$, and $\gamma$ quantifies how significant a single request is when compared to the sum total of all requests. We discuss implications of our results to several special cases including online combinatorial auctions, network routing and the adwords problem.