Improved Submodular Secretary Problem with Shortlists
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First, for the for the submodular k-secretary problem with shortlists [1], we provide a near optimal 1-1/e-ϵ approximation using shortlist of size O(k poly(1/ϵ)). In particular, we improve the size of shortlist used in <cit.> from O(k 2^poly(1/ϵ)) to O(k poly(1/ϵ)). 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/ϵ)). It exponentially improves the running time and memory of <cit.> in terms of 1/ϵ. 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 matroid secretary problem <cit.>, in which the algorithm is allowed to have a shortlist. We design an algorithm that achieves a 1/2(1-1/e^2 -ϵ) competitive ratio for any constant ϵ>0, using a shortlist of size O(k poly(1/ϵ)). Moreover, we generalize our results to the case of p-matchoid constraints and give a 1/p+1(1-1/e^p+1-ϵ ) approximation using shortlist of size O(k poly(1/ϵ)). It asymptotically approaches the best known offline guarantee 1/p+1 [22]. Furthermore, we show that our algorithms can be implemented in the streaming setting using O(k poly(1/ϵ)) memory.
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