An Improved Lower Bound for Matroid Intersection Prophet Inequalities

TL;DR

This paper improves the lower bound for prophet inequalities under matroid intersection constraints, showing that the difficulty of the problem grows faster than previously established, especially for intersection of partition matroids.

Contribution

It establishes a stronger lower bound of q^{1/2+Ω(1/ log log q)} for prophet inequalities, refining the understanding of their complexity.

Findings

New lower bound surpasses previous ((((q)))) approximation.

Uses advanced combinatorial techniques to relate graph product dimension to matroid intersection complexity.

Improves theoretical understanding of prophet inequalities in combinatorial optimization.

Abstract

We consider prophet inequalities subject to feasibility constraints that are the intersection of $q$ matroids. The best-known algorithms achieve a $\Theta(q)$-approximation, even when restricted to instances that are the intersection of $q$ partition matroids, and with i.i.d.~Bernoulli random variables. The previous best-known lower bound is $\Theta(\sqrt{q})$ due to a simple construction of [Kleinberg-Weinberg STOC 2012] (which uses i.i.d.~Bernoulli random variables, and writes the construction as the intersection of partition matroids). We establish an improved lower bound of $q^{1/2+\Omega(1/\log \log q)}$ by writing the construction of [Kleinberg-Weinberg STOC 2012] as the intersection of asymptotically fewer partition matroids. We accomplish this via an improved upper bound on the product dimension of a graph with $p^p$ disjoint cliques of size $p$, using recent techniques developed in [Alon-Alweiss European Journal of Combinatorics 2020].