Non-Adaptive Prophet Inequalities for Minor-Closed Classes of Matroids

TL;DR

This work studies constant-competitive non-adaptive mechanisms for the matroid prophet inequality, a problem where a non-adaptive online selector must pick an independent set under known distributions. It develops an ex-ante relaxation framework via the matroid polytope and leverages tree-decomposition and field-representability to extend constant-competitive guarantees from graphic and cographic matroids to broader minor-closed families, including $k$-column sparse and $oldsymbol{\eta}$-sparse classes. Key results include a $16$-competitive mechanism for simple graphic matroids, a $(2^{k+2}k)$-competitive mechanism for $k$-column sparse matroids, a $6$-competitive mechanism for cographic matroids, and 256-competitive guarantees for regular matroids under a Structural Hypothesis, with minor-closed representable families handled via projections, lifts, and decompositions. The findings significantly broaden the applicability of non-adaptive prophet inequalities and connect matroid theory with online decision-making under uncertainty, enabling robust, single-threshold strategies across diverse combinatorial constraints. The methods enhance understanding of when non-adaptive (static-threshold) rules can rival adaptive strategies, with potential implications for online selection under complex feasibility constraints.

Abstract

We consider the matroid prophet inequality problem. This problem has been extensively studied in the case of adaptive mechanisms. In particular, there is a tight $2$-competitive mechanism for all matroids. However, it is not known what classes of matroids admit non-adaptive mechanisms with constant guarantee. Recently, it was shown that there are constant-competitive non-adaptive mechanisms for graphic matroids. In this work, we show that various known classes of matroids admit constant-competitive non-adaptive mechanisms.

Theorems & Definitions (64)

• Theorem 1: Uniform Rank $1$ Matroid samuelcahn1984comparison
• Theorem 2: Graphic Matroid chawla2020non
• Theorem 3
• Theorem 4: $k$-Column Sparse Matroids
• Theorem 5: Cographic Matroids
• Theorem 6: $\gamma$-Sparse Matroids soto2013matroid
• Theorem 7: Regular Matroids
• Theorem 8
• Theorem 9: Laminar Matroid azar
• Theorem 10: Truncated Partition Matroid caramanis