Prophet Inequalities on the Intersection of a Matroid and a Graph

We consider prophet inequalities in a setting where agents correspond to both elements in a matroid and vertices in a graph. A set of agents is feasible if they form both an independent set in the matroid and an independent set in the graph. Our main result is an ex-ante 1/(2d+2)-prophet inequality, where d is a graph parameter upper-bounded by the maximum size of an independent set in the neighborhood of any vertex. We establish this result through a framework that sets both dynamic prices for elements in the matroid (using the method of balanced thresholds), and static but discriminatory prices for vertices in the graph (motivated by recent developments in approximate dynamic programming). The threshold for accepting an agent is then the sum of these two prices. We show that for graphs induced by a certain family of interval-scheduling constraints, the value of d is 1. Our framework thus provides the first constant-factor prophet inequality when there are both matroid-independence constraints and interval-scheduling constraints. It also unifies and improves several results from the literature, leading to a 1/2-prophet inequality when agents have XOS valuation functions over a set of items and use them for a finite interval duration, and more generally, a 1/(d+1)-prophet inequality when these items each require a bundle of d resources to procure.