Approximating Optimum Online for Capacitated Resource Allocation

TL;DR

This paper introduces a polynomial-time online algorithm for capacitated resource allocation that surpasses the traditional 1/2-approximate barrier, achieving a 0.5115-approximation to the optimal online solution, with implications for online Bayesian problems.

Contribution

The paper presents the first polynomial-time algorithm exceeding the 1/2-approximate barrier for online capacitated resource allocation, using a novel LP rounding technique and addressing positive correlation challenges.

Findings

Achieves a (1/2 + 0.0115)-approximation to the optimal online algorithm.

Introduces a two-proposal LP rounding method for capacitated online matching.

Highlights the importance of problem-specific insights in online algorithm benchmarks.

Abstract

We study online capacitated resource allocation, a natural generalization of online stochastic max-weight bipartite matching. This problem is motivated by ride-sharing and Internet advertising applications, where online arrivals may have the capacity to serve multiple offline users. Our main result is a polynomial-time online algorithm which is $(1/2 + \kappa)$-approximate to the optimal online algorithm for $\kappa = 0.0115$. This can be contrasted to the (tight) $1/2$-competitive algorithms to the optimum offline benchmark from the prophet inequality literature. Optimum online is a recently popular benchmark for online Bayesian problems which can use unbounded computation, but not "prophetic" knowledge of future inputs. Our algorithm (which also works for the case of stochastic rewards) rounds a generalized LP relaxation from the unit-capacity case via a two-proposal algorithm, as in previous works in the online matching literature. A key technical challenge in deriving our guarantee is bounding the positive correlation among users introduced when rounding our LP relaxation online. Unlike in the case of unit capacities, this positive correlation is unavoidable for guarantees beyond $1/2$. Conceptually, our results show that the study of optimum online as a benchmark can reveal problem-specific insights that are irrelevant to competitive analysis.