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

2026-07-22

AI Summary

This work addresses online decision-making under matroid constraints by proposing a nearly non-adaptive algorithm. The approach constructs a new matroid as the direct sum of multiple constituent matroids and assigns a single pre-specified threshold to each component. Upon the arrival of each element, the algorithm makes irrevocable decisions solely based on these fixed thresholds and feasibility within the newly constructed matroid. Remarkably, this method achieves an optimal $1/2$-approximation ratio relative to a stronger ex-ante relaxation, without relying on adaptive thresholding. This result significantly improves upon the previous best guarantee of $1/4$ for similar settings and matches the theoretical upper bound, thereby establishing a new benchmark for near-optimal performance in non-adaptive online selection under matroid constraints.

Abstract

Prophet inequalities are a fundamental model for online decision-making under uncertainty. For matroid constraints, Kleinberg and Weinberg gave a tight $1/2$-approximation, but their algorithm uses adaptive thresholds that depend on the previously accepted elements. Specifically, the mechanism of Kleinberg and Weinberg accepts an arriving element if and only if it is feasible to add with respect to the matroid constraint and the value of the arriving element passes its threshold; but this threshold depends on the elements accepted so far and on the arriving element itself. Later, Feldman, Svensson, and Zenklusen showed that one can give a $1/4$-approximation for general matroids. Their algorithm is almost non-adaptive, i.e., it uses non-adaptive thresholds but changes the underlying matroid to another ``stricter'' matroid. Feldman, Svensson, and Zenklusen also showed that no constant approximation is possible in general matroids using non-adaptive thresholds if one does not change the underlying matroid. We give a new almost non-adaptive algorithm for matroid prophet inequalities that achieves a $1/2$ guarantee. We change the underlying matroid to a ``stricter'' new matroid that is a direct sum of several matroids. For each part of the new matroid, we precompute a single non-adaptive threshold. Once the elements start to arrive, we accept an arriving element as long as it is feasible with respect to the new ``stricter'' matroid and its value passes the precomputed threshold. In addition, we guarantee that our algorithm achieves the $1/2$ guarantee not simply with respect to the prophet's expected gain, but with respect to the stronger ex-ante relaxation value. Thus, we provide the first almost non-adaptive algorithm for the matroid prophet inequality that achieves the best-possible approximation guarantee of $1/2$.

Problem

• prophet inequalities
• matroid constraints
• non-adaptive thresholds
• online decision-making
• approximation algorithms

Innovation

• matroid prophet inequalities
• almost non-adaptive
• 1/2-approximation
• ex-ante relaxation
• threshold algorithms