Submodular Secretary Problems: Cardinality, Matching, and Linear Constraints

  title={Submodular Secretary Problems: Cardinality, Matching, and Linear Constraints},
  author={Thomas Kesselheim and Andreas T{\"o}nnis},
We study various generalizations of the secretary problem with submodular objective functions. Generally, a set of requests is revealed step-by-step to an algorithm in random order. For each request, one option has to be selected so as to maximize a monotone submodular function while ensuring feasibility. For our results, we assume that we are given an offline algorithm computing an $\alpha$-approximation for the respective problem. This way, we separate computational limitations from the ones… 

The submodular secretary problem under a cardinality constraint and with limited resources

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Submodular Matroid Secretary Problem with Shortlists

An algorithm is designed that achieves a $\frac{1}{2}(1-1/e^2-\epsilon-O(1/k)))$ competitive ratio for any constant $\epsil on>0$, using a shortlist of size $O(k)$.

Improved Submodular Secretary Problem with Shortlists

A near optimal approximation algorithm for random-order streaming of monotone submodular functions under cardinality constraints, using memory $O(k poly(1/\epsilon), which exponentially improves the running time and memory of \cite{us} in terms of $1/£1 and asymptotically approaches the best known offline guarantee $\frac{1}{p+1}$.

Combinatorial Secretary Problems with Ordinal Information

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Matroid Secretary Problems

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New Results for the k-Secretary Problem

A natural deterministic algorithm designed to have competitive ratios strictly greater than 1/e for small k >= 2, which is hardly more complex than the elegant strategy for the classical secretary problem, optimal for k=1, and works for all k >= 1.

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Budget-Feasible Mechanism Design for Non-Monotone Submodular Objectives: Offline and Online

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Competitive Algorithms for Online Budget-Constrained Continuous DR-Submodular Problems

The first bound on the competitive ratio of online monotone DR-submodular function maximization subject to linear packing constraints is obtained, which matches the known tight bound in the special case of linear objective function.

Incentives in dynamic markets

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Constrained Non-monotone Submodular Maximization: Offline and Secretary Algorithms

These ideas are extended to give a simple greedy-based constant factor algorithms for non-monotone submodular maximization subject to a knapsack constraint, and for (online) secretary setting subject to uniform matroid or a partition matroid constraint.

The Simulated Greedy Algorithm for Several Submodular Matroid Secretary Problems

This paper focuses on the case that the valuation function is a non-negative and monotonically non-decreasing submodular function and introduces a general algorithm for such sub modular matroid secretary problems and obtains constant competitive algorithms for the cases of laminarMatroids and transversal matroids.

Submodular secretary problem and extensions

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Improved Competitive Ratios for Submodular Secretary Problems (Extended Abstract)

A simple observation which states that the random order of the input can be generated by independently choosing a random continuous arrival time for each secretary enables us to improve the competitive ratio of several known and studied variants of the secretary problem.

Approximations for Monotone and Nonmonotone Submodular Maximization with Knapsack Constraints

This paper establishes a strong relation between the discrete problem and its continuous relaxation, obtained through extension by expectation of the submodular function, and shows that the probabilistic domain defined by a continuous solution can be reduced to yield a polynomial-size domain.

Matroids, secretary problems, and online mechanisms

An O(log k)-competitive algorithm for general matroids (where k is the rank of the matroid), and constant-competitive algorithms for several special cases including graphicMatroids, truncated partition matroIDS, and bounded degree transversal matroid algorithms are presented.

Submodular Maximization with Cardinality Constraints

Improved approximations for two variants of the cardinality constraint for non-monotone functions are presented and a simple randomized greedy approach is presented where in each step a random element is chosen from a set of "reasonably good" elements.

The Submodular Secretary Problem Goes Linear

  • Moran FeldmanR. Zenklusen
  • Computer Science, Mathematics
    2015 IEEE 56th Annual Symposium on Foundations of Computer Science
  • 2015
It is shown that any O(1)-competitive algorithm for MSP, even restricted to a particular matroid class, can be transformed in a black-box way to an O( 1)- competitive algorithm for SMSP over the same matroidclass, which implies that SMSP is not harder than MSP.

Online Submodular Welfare Maximization: Greedy Beats 1/2 in Random Order

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Best Algorithms for Approximating the Maximum of a Submodular Set Function

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