Stochastic Online Correlated Selection
Ziyun Chen, Zhiyi Huang, Enze Sun
TL;DR
This work introduces Stochastic Online Correlated Selection (SOCS), a unified online rounding framework for non-IID stochastic online submodular welfare maximization and several canonical problems (Online Stochastic/Display Ads, AdWords, and Query-Commit). It centers on a convergence-rate metric g(y) and a Type Decomposition technique that reduces general SOCS to a two-way surrogate setting, preserving expected allocations while maximizing two-way opportunities. The authors derive optimal two-way SOCS for unweighted/vertex-weighted matching, and extend results to AdWords and Display Ads, breaking longstanding barriers such as the 1−1/e benchmark in the non-IID stochastic model and delivering the first multi-way OCS for AdWords. Through LP relaxations, negative association tools, and new analytical recurrences, the paper achieves state-of-the-art competitive ratios: 0.69 for Non-IID Online Stochastic Matching, 0.705 in Random-Order and Query-Commit models, 0.6338 for Stochastic AdWords, 0.644 for Stochastic Display Ads, and a 0.504 competitive ratio for AdWords in the adversarial model. The results demonstrate a versatile, modular framework with broad applicability and provide substantial avenues for genuine multi-way SOCS and tightened convergence-rate analyses across online submodular welfare problems.
Abstract
We study Stochastic Online Correlated Selection (SOCS), a family of online rounding algorithms for Non-IID Stochastic Online Submodular Welfare Maximization and special cases such as Online Stochastic Matching, Stochastic AdWords, and Stochastic Display Ads. At each step, the algorithm sees an online item's type and fractional allocation, then immediately allocates it to an agent. We propose a metric called the convergence rate for the quality of SOCS. This is cleaner than most metrics in the OCS literature. We propose a Type Decomposition that reduces SOCS to the two-way special case. First, we sample a surrogate type with half-integer allocation. The rounding is trivial for a one-way type fully allocated to an agent. For a two-way type split equally between two agents, we round it using two-way SOCS. We design the distribution of surrogate types to get two-way types as often as possible while respecting the original fractional allocation in expectation. Following this framework, we make progress on numerous problems: 1) Online Stochastic Matching: We improve the state-of-the-art $0.666$ competitive ratio for unweighted/vertex-weighted matching to $0.69$. 2) Query-Commit Matching: We enhance the ratio to $0.705$ in the Query-Commit model, improving the best previous $0.696$ and $0.662$ for unweighted and vertex-weighted matching. 3) Stochastic AdWords: We give a $0.6338$ competitive algorithm, breaking the $1-\frac{1}{e}$ barrier and answering a decade-old open question. 4) AdWords: The framework applies to the adversarial model if the rounding is oblivious to future items' distributions. We get the first multi-way OCS for AdWords, addressing an open question about OCS. This gives a $0.504$ competitive ratio for AdWords, improving the previous $0.501$. 5) Stochastic Display Ads: We design a $0.644$ competitive algorithm, breaking the $1-\frac{1}{e}$ barrier.