# Submodular Secretary Problems: Cardinality, Matching, and Linear Constraints

@inproceedings{Kesselheim2017SubmodularSP,
title={Submodular Secretary Problems: Cardinality, Matching, and Linear Constraints},
author={Thomas Kesselheim and Andreas T{\"o}nnis},
booktitle={APPROX-RANDOM},
year={2017}
}
• Published in APPROX-RANDOM 1 July 2016
• Computer Science, Mathematics
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…
15 Citations

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

• Computer Science
ArXiv
• 2017
This work proposes a $0.1933$-competitive anytime algorithm, which performs only a single evaluation of the marginal contribution for each observed item, and requires a memory of order only $k$ (up to logarithmic factors), where k is the cardinality constraint.

### 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/Â£1 and asymptotically approaches the best known offline guarantee $\frac{1}{p+1}$.

### Combinatorial Secretary Problems with Ordinal Information

• Computer Science
ICALP
• 2017
This paper initiates the study of combinatorial secretary problems with ordinal information, in which the decision maker only needs to be aware of a preference order consistent with the values of arrived elements, and provides a lower bound of $\Omega(\sqrt{n}/(\log n))$ for algorithms that are oblivious to the matroid structure.

### Matroid Secretary Problems

• Mathematics, Computer Science
J. ACM
• 2018
This work defines a generalization of the classical secretary problem called the matroid secretary problem, and presents an O(log k)-competitive algorithm for general matroids, and constant-competitive algorithms for several special cases including graphic matroid, truncated partition matroid, and bounded degree transversal matroid.

### New Results for the k-Secretary Problem

• Computer Science, Mathematics
ISAAC
• 2019
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.

### Packing returning secretaries

• Computer Science, Mathematics
ISAAC
• 2018
This work focuses on minimizing the expected number of postponements when computing an optimal solution to the online bipartite matching problem and shows a tight bound of O(r′ log(n/r′), where r′ is the minimum rank of the matroid and the dual matroid.

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

• Computer Science
EC
• 2019
This work focuses on the case of general (non-monotone) submodular valuation functions and derives the first truthful, budget-feasible and O(1)-approximation mechanisms that run in polynomial time in the value query model, for both offline and online auctions.

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

• Computer Science, Mathematics
ArXiv
• 2019
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

This thesis studies extensions of the so-called smoothness framework for mechanisms, a very useful technique for bounding the inefficiency of equilibria, to the cases of varying mechanism availability and participation of risk-averse players, and designs new algorithms that obtain constant competitive ratios for a variety of combinatorial problems.

## References

SHOWING 1-10 OF 32 REFERENCES

### Constrained Non-monotone Submodular Maximization: Offline and Secretary Algorithms

• Mathematics, Computer Science
WINE
• 2010
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

• Mathematics, Computer Science
Theory of Computing Systems
• 2015
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

• Mathematics, Computer Science
TALG
• 2013
This article considers a very general setting of the classic secretary problem, in which the goal is to select k secretaries so as to maximize the expectation of a submodular function which defines efficiency of the selected secretarial group based on their overlapping skills, and presents the first constant-competitive algorithm for this case.

### Improved Competitive Ratios for Submodular Secretary Problems (Extended Abstract)

• Mathematics, Computer Science
APPROX-RANDOM
• 2011
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

• Mathematics, Computer Science
Math. Oper. Res.
• 2013
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

• Mathematics, Computer Science
SODA '07
• 2007
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

• Computer Science
SODA
• 2014
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

• 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

• Computer Science
STOC
• 2015
This paper solves the online version of Submodular Welfare Maximization (SWM) by demonstrating that the greedy algorithm has a competitive ratio of at least 0.505 for online SWM in the random order model, and defines the classes of second-order modular, supermodular, and submodular functions, which are likely to be of independent interest in sub modular optimization.

### Best Algorithms for Approximating the Maximum of a Submodular Set Function

• Mathematics
Math. Oper. Res.
• 1978
This work presents a family of algorithms that involve the partial enumeration of all sets of cardinality q and then a greedy selection of the remaining elements, q = 0,..., K-1.