| Popis: |
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. |
