New Math Breakthrough Could Make Online Auctions Fairer and More Efficient

The article reports resolving the matroid secretary conjecture in its strong form and the single-sample matroid prophet inequality problem, achieving competitive ratios e and 2, respectively. Both ratios are optimal, even for rank-one matroids. For matroid secretary, the ordinal algorithm accepts the elements of the canonical optimum independently, each with probability 1/e; it needs only the number of elements in advance and uses an expected polynomial number of independence queries on arrived elements.

For single-sample matroid prophet inequalities, the algorithm uses one sample per independent value distribution and O(n^2) independence queries on a known n-element matroid. It works for every arrival order fixed independently of the samples, values, and algorithmic randomness. The selected elements and their values have exactly the joint distribution obtained by independently retaining each element-value pair in the canonical optimum of a fresh product draw with probability 1/2.

The main algorithms maintain a reference state whose greedy solution contains every accepted element; updates, made even after rejections, preserve exact distributional invariants. A short reversible-update argument gives a 4-probability-competitive secretary algorithm, already resolving the original conjecture; a related invariant gives the optimal single-sample prophet inequality. For secretary, the article characterizes the distributions achievable by ordinal algorithms through a finite polytope, and a continuous-time formulation lets the authors design an accepted-set distribution achieving the optimal ratio e and realize it efficiently. The article also obtains guarantees for matroid intersections.