Submodular Matroid Secretary Problem with Shortlists

Abstract

In the matroid secretary problem, the elements of a matroid $\mathcal{M}$ arrive in random order. Once we observe an item we need to irrevocably decide whether or not to accept it. The set of selected elements should form an independent set of the matroid. The goal is to maximize the total sum of the values assigned to these elements. We introduce a version of this problem motivated by the shortlist model in [Agrawal et al.]. In this setting, the algorithm is allowed to choose a subset of items as part of a shortlist, possibly more than $k=rk(\mathcal{M})$ items. Then, after seeing the entire input, the algorithm can choose an independent subset from the shortlist. Furthermore we generalize the objective function to any monotone submodular function. Is there an online algorithm achieve a constant competitive ratio using a shortlist of size $O(k)$? We design an algorithm that achieves a $\frac{1}{2}(1-1/e^2-ε-O(1/k))$ competitive ratio for any constant $ε>0$, using a shortlist of size $O(k)$. This is especially surprising considering that the best known competitive ratio for the matroid secretary problem is $O(\log \log k)$. An important application of our algorithm is for the random order streaming of submodular functions. We show that our algorithm can be implemented in the streaming setting using $O(k)$ memory. It achieves a $\frac{1}{2}(1-1/e^2-ε-O(1/k))$ approximation. The previously best known approximation ratio for streaming submodular maximization under matroid constraint is 0.25 (adversarial order) due to [Feldman et al.], [Chekuri et al.], and [Chakrabarti et al.]. Moreover, we generalize our results to the case of p-matchoid constraints and give a $\frac{1}{p+1}(1-1/e^{p+1}-ε-O(1/k))$ approximation using $O(k)$ memory, which asymptotically approaches the best known offline guarantee $\frac{1}{p+1}$ [Nemhauser et al.]