AI 中文总结
研究拟阵先知不等式，提出新的几乎非自适应算法，通过将拟阵改为新的“更严格”拟阵并预计算非自适应阈值，实现了1/2近似保证，且在更强的事前松弛值上也能达到，是首个达最优近似保证的此类算法。
AI 中文摘要
先知不等式是不确定性下在线决策的基本模型。对于拟阵约束，Kleinberg和Weinberg给出了精确的1/2近似，但他们的算法使用依赖于先前接受元素的自适应阈值。Feldman等人给出了1/4近似且算法几乎非自适应，但不改变拟阵时一般拟阵无法用非自适应阈值得到常数近似。我们给出了新的几乎非自适应算法，将拟阵改为由几个拟阵直和而成的“更严格”新拟阵，为各部分预计算单个非自适应阈值，只要元素对新拟阵可行且值超过阈值就接受。此外，该算法不仅在先知期望收益上达到1/2保证，在更强的事前松弛值上也能达到，是首个达到最优1/2近似保证的几乎非自适应算法。
英文摘要
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.
