Approximating Optimum Online for Capacitated Resource Allocation

TL;DR

This work addresses online capacitated resource allocation, where online resources with capacities must be assigned to offline users to maximize expected social welfare, and benchmarks against the computationally unbounded online optimum rather than the offline prophet. It introduces a polynomial-time algorithm that rounds a generalized LP online via a two-proposal pivotal-sampling scheme, achieving a guaranteed $0.5 + \kappa$-approximation to the optimal online welfare with $\kappa = 0.0115$, and extends to stochastic rewards. A key technical contribution is a careful control of positive correlation among offline user availabilities across two rounds, plus a detailed inductive analysis that bounds complex dependencies while preserving near-optimal edge probabilities. The results demonstrate that the optimum-online benchmark yields problem-specific insights, and that such beat-a-half guarantees can be achieved in capacitated settings, with implications for online auctions, mechanism design, and related online matching problems. The paper also provides a polynomial-time sampling-based variant and outlines extensions to non-Bernoulli arrivals and general reward distributions, highlighting the practical relevance for large-scale, stochastic online decision problems.

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 $(1/2 + κ)$-approximate to the optimal online algorithm for $κ= 0.0115$. 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.

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1: as in srinivasan2001distributions
• proof
• proof
• Theorem 4.1
• proof
• Lemma 4.4
• proof : Proof of Theorem \ref{['theorem:main_theorem']}.