Simple and Almost Non-Adaptive \(\frac{1}{2}\)-Approximation for Matroid Prophet Inequalities

Abstract

Prophet inequalities are a fundamental model for online decision-making under uncertainty. For matroid constraints, Kleinberg and Weinberg gave a tight $\frac{1}{2}$-approximation using adaptive thresholds, while Feldman, Svensson, and Zenklusen obtained a $\frac{1}{4}$-approximation via an online contention resolution scheme (OCRS). We give the first almost non-adaptive algorithm for general matroid prophet inequalities achieving the optimal $\frac{1}{2}$ guarantee, in fact with respect to the stronger ex-ante relaxation. Starting from an optimal ex-ante solution $x$, we reduce to a Bernoulli instance, replace the original matroid by a stricter direct sum of minors, and assign fixed thresholds to the resulting components. Translating the rule back to the original distributions, an element $e$ can be accepted only when its realized value lies in its top $x_e$-quantile and adding it preserves the corresponding stricter matroid constraint. We also give a second almost non-adaptive $\frac{1}{2}$-approximation based on a different threshold rule. This formulation extends naturally to intersections of matroids and yields an almost non-adaptive $(q+1)$-approximation for prophet inequalities under the intersection of $q$ arbitrary matroids, again with respect to the ex-ante relaxation. This matches the previously known $(q+1)$ guarantee for intersections of $q$ partition matroids, due to Alon, Pollner, and Weinberg, while extending it to arbitrary matroids. For the intersection result, each matroid is replaced by a stricter direct sum of minors, and a common surplus vector determines fixed element thresholds across all $q$ constraints. We prove the existence of such a vector using Brouwer's fixed-point theorem and give a polynomial-time procedure to compute it.