Prophet Secretary for k-Knapsack and l-Matroid Intersection via Continuous Exchange Property

We study the k-knapsack and l-matroid constrained prophet secretary problem, which is a combinatorial prophet secretary problem whose feasible domain is the intersection of k-knapsack constraints and l-matroid constraints. Here, the prices of the items and the structure of the matroids are deterministic and known in advance, and the values of the items are stochastic and their distributions are known in advance. We derive a constant-factor approximation algorithm for this problem. We adapt Ehsani et al. (2018)'s technique for the matroid constraint to the knapsack constraint via continuous relaxation. For this purpose, we prove an "exchange property" of the knapsack constraint.