Papers for
online market designers
Papers whose findings have a practical use for this group, as judged from the abstract. Open a paper to read what it means in practice.
Submodular secretary problem achieves optimal competitive ratios under constraints
The Submodular Secretary Problem: Tight $1-1/e$ for Monotone Objectives, and Strictly Between $1/e$ and $1/2$ in General
Abstract: In the submodular secretary problem with a cardinality constraint, $n$ elements arrive in uniformly random order, and an algorithm must irrevocably accept or reject each one on arrival so as to maximize a nonnegative submodular function over sets of at most $k$ elements. For monotone objectives we settle the problem: the optimal competitive ratio tends to exactly $1-1/e$ as $k\to\infty$, and its worst case over all budgets $k$ is exactly $1/e$. The bound $1-1/e$ cannot be exceeded even with unlimited computation, and it is achieved by a single algorithm with at most $4n+1$ value queries; previously the best ratio without a shortlist was $1/e-o(1)$, with exponentially many queries. For general objectives, where the best known ratio was about $0.107$, we show that the optimal limiting ratio lies strictly between $1/e$ and $1/2$: we achieve $1/e+3.5\cdot10^{-6}$ for large $k$, and no algorithm achieves more than $0.497053$. We also achieve $0.2677$ at every budget. The main new ingredient of our algorithm is eligibility attenuation: an element that passes the price test at time $t$ is accepted with probability $e^{-τ}$, where $τ$ is the total earlier time during which it would have passed, which makes the probability that it has already been selected exactly $1-e^{-τ}$.
General reduction improves online allocation with limited samples
Single-Sample Prophet Inequalities: A Combinatorial to Single-Item Reduction
Abstract: We study single-sample prophet inequalities for online combinatorial allocation. Our main contribution is a general reduction from combinatorial to single-item prophet inequalities for valuation classes admitting suitable supporting prices. The reduction uses a free-disposal value to separate buyer-side combinatorial constraints from item-side supply constraints, yielding a modular framework that applies in the stronger Game of Googol model. This framework yields a $\frac{1}{6\sqrt{3}}\approx\frac{1}{10.4}$-competitive single-sample prophet inequality and a $(β_{k-1}/4)$-competitive $k$-sample prophet inequality for XOS valuations, where $β_k$ is the competitive ratio of a $k$-sample single-item prophet inequality, improving upon the work of [DKL+24]. Both results extend directly to divisible resources with capped-XOS valuations. Along the way, we obtain new results for online free disposal and an optimal single-sample prophet inequality for fractional knapsack in the Game of Googol model.