As Easy as Rank One: Optimal Algorithms for Matroid Secretary and Single-Sample Matroid Prophet
We resolve 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, our 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, our 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 $1/2$. Our main algorithms maintain a reference state whose greedy solution contains every accepted element. Its 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, we characterize the distributions achievable by ordinal algorithms through a finite polytope. A continuous-time formulation then lets us design an accepted-set distribution achieving the optimal ratio $e$ and realize it efficiently. We also obtain guarantees for matroid intersections.