Improved Submodular Secretary Problem with Shortlists

TL;DR

This paper improves approximation algorithms for the submodular secretary problem with shortlists, reducing shortlist size and extending results to matroid constraints, with applications to streaming algorithms.

Contribution

It introduces near optimal algorithms with smaller shortlists for submodular secretary problems under various constraints, including matroids, and extends to streaming settings.

Findings

Achieves near optimal $1-1/e-psilon$ approximation with smaller shortlists.

Provides a $rac{1}{2}(1-1/e^2-psilon)$ competitive ratio for matroid constraints.

Extends results to $p$-matchoid constraints with asymptotic optimality.

Abstract

First, for the for the submodular $k$-secretary problem with shortlists [1], we provide a near optimal $1-1/e-\epsilon$ approximation using shortlist of size $O(k poly(1/\epsilon))$. In particular, we improve the size of shortlist used in \cite{us} from $O(k 2^{poly(1/\epsilon)})$ to $O(k poly(1/\epsilon))$. As a result, we provide a near optimal approximation algorithm for random-order streaming of monotone submodular functions under cardinality constraints, using memory $O(k poly(1/\epsilon))$. It exponentially improves the running time and memory of \cite{us} in terms of $1/\epsilon$. Next we generalize the problem to matroid constraints, which we refer to as submodular matroid secretary problem with shortlists. It is a variant of the \textit{matroid secretary problem} \cite{feldman2014simple}, in which the algorithm is allowed to have a shortlist. We design an algorithm that achieves a $\frac{1}{2}(1-1/e^2 -\epsilon)$ competitive ratio for any constant $\epsilon>0$, using a shortlist of size $O(k poly(\frac{1}{\epsilon}))$. Moreover, we generalize our results to the case of $p$-matchoid constraints and give a $\frac{1}{p+1}(1-1/e^{p+1}-\epsilon )$ approximation using shortlist of size $O(k poly(\frac{1}{\epsilon}))$. It asymptotically approaches the best known offline guarantee $\frac{1}{p+1}$ [22]. Furthermore, we show that our algorithms can be implemented in the streaming setting using $O(k poly(\frac{1}{\epsilon}))$ memory.