Saar Cohen, Nicholas Teh, Paul W. Goldberg +1cs.GT cs.AI cs.LG cs.MA econ.TH
We study an online variant of discrete fair division under generalized assignment budget constraints. Goods arrive one at a time and must be assigned irrevocably to a feasible agent or to charity, which holds all unallocated goods, while fairness is evaluated only against budget-feasible subsets of every recipient's bundle. We first show that, without additional structure, no deterministic online algorithm can guarantee any fixed approximation to feasible envy-freeness, even in highly symmetric instances. We then identify bounded density spread as a structural condition that restores meaningful guarantees, obtaining approximation algorithms for arbitrary item sizes and showing that, under common valuations and sufficiently small goods, these guarantees can be strengthened to an optimal deterministic frontier. We further study resource augmentation, where the online algorithm is allowed slightly larger budgets than the fairness benchmark, and characterize the resulting improvement in the achievable guarantees. Finally, we develop a learning-augmented framework based on predicting joint value-size types, proving consistency under perfect predictions, robustness to prediction error, and showing that separate predictions of value and size marginals are insufficient to recover strong fairness guarantees.
In learning-augmented online algorithms, predictions are usually valued for what they say: a value estimate, a solution, or an algorithmic recommendation. This paper shows that predictions can also be valuable solely due to their arrival time. We study the fundamental secretary problem augmented with a stochastic precursor: a content-free signal that is guaranteed to arrive no later than the best item, but is otherwise stochastically timed. The signal does not carry any additional information; nevertheless, its timing alone changes the structure of optimal stopping. We characterize optimal policies in the random-order and adversarial-order models. In random order, a single uniformly timed precursor already gives success probability at least $\frac12$, improving on the classic $\frac1e$ benchmark. With increasingly late precursors, the success probability approaches $1$. In adversarial order, for which traditional models do not admit strong guarantees, sufficiently concentrated precursors recover constant success guarantees. Our results show that such novel forms of asynchronous temporal information are a distinct and powerful form of advice in online decision making and may also be effective for other problems.