research-article

Submodular secretary problem and extensions

Authors:

Mohammadhossein Bateni,

Mohammadtaghi Hajiaghayi,

Morteza ZadimoghaddamAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 9, Issue 4

Article No.: 32, Pages 1 - 23

https://doi.org/10.1145/2500121

Published: 03 October 2013 Publication History

Abstract

Online auction is the essence of many modern markets, particularly networked markets, in which information about goods, agents, and outcomes is revealed over a period of time, and the agents must make irrevocable decisions without knowing future information. Optimal stopping theory, especially the classic secretary problem, is a powerful tool for analyzing such online scenarios which generally require optimizing an objective function over the input. The secretary problem and its generalization the multiple-choice secretary problem were under a thorough study in the literature. In this article, we consider a very general setting of the latter problem called the submodular secretary problem, in which the goal is to select k secretaries so as to maximize the expectation of a (not necessarily monotone) submodular function which defines efficiency of the selected secretarial group based on their overlapping skills. We present the first constant-competitive algorithm for this case. In a more general setting in which selected secretaries should form an independent (feasible) set in each of l given matroids as well, we obtain an O(l log² r)-competitive algorithm generalizing several previous results, where r is the maximum rank of the matroids. Another generalization is to consider l knapsack constraints (i.e., a knapsack constraint assigns a nonnegative cost to each secretary, and requires that the total cost of all the secretaries employed be no more than a budget value) instead of the matroid constraints, for which we present an O(l)-competitive algorithm. In a sharp contrast, we show for a more general setting of subadditive secretary problem, there is no õ(√n)-competitive algorithm and thus submodular functions are the most general functions to consider for constant-competitiveness in our setting. We complement this result by giving a matching O(√n)-competitive algorithm for the subadditive case. At the end, we consider some special cases of our general setting as well.

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Buchbinder NFeldman MMohar BShinkar IO'Donnell R(2024)Constrained Submodular Maximization via New Bounds for DR-Submodular FunctionsProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649630(1820-1831)Online publication date: 10-Jun-2024
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Banihashem KBiabani LGoudarzi SHajiaghayi MJabbarzade PMonemizadeh MOh ANaumann TGloberson ASaenko KHardt MLevine S(2023)Dynamic non-monotone submodular maximizationProceedings of the 37th International Conference on Neural Information Processing Systems10.5555/3666122.3666883(17369-17382)Online publication date: 10-Dec-2023
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Index Terms

Submodular secretary problem and extensions
1. Applied computing
  1. Law, social and behavioral sciences
    1. Economics
2. Theory of computation
  1. Design and analysis of algorithms

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cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 9, Issue 4

September 2013

131 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/2533288

Issue’s Table of Contents

Copyright © 2013 ACM.

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Publication History

Published: 03 October 2013

Accepted: 01 February 2013

Revised: 01 October 2012

Received: 01 November 2010

Published in TALG Volume 9, Issue 4

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Division of Computing and Communication Foundations
University of Maryland Research and Scholarship Award (RASA)
National Science Foundation
Office of Naval Research
Air Force Office of Scientific Research
Defense Advanced Research Projects Agency

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Cited By

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https://doi.org/10.1287/moor.2023.0210
Buchbinder NFeldman MMohar BShinkar IO'Donnell R(2024)Constrained Submodular Maximization via New Bounds for DR-Submodular FunctionsProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649630(1820-1831)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649630
Banihashem KBiabani LGoudarzi SHajiaghayi MJabbarzade PMonemizadeh MOh ANaumann TGloberson ASaenko KHardt MLevine S(2023)Dynamic non-monotone submodular maximizationProceedings of the 37th International Conference on Neural Information Processing Systems10.5555/3666122.3666883(17369-17382)Online publication date: 10-Dec-2023
https://dl.acm.org/doi/10.5555/3666122.3666883
Yang RGao SHan LLi GZhao Z(2023)Approximating ($m_{B},m_{P}$)-Monotone BP Maximization and ExtensionsTsinghua Science and Technology10.26599/TST.2022.901003328:5(906-915)Online publication date: Oct-2023
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Gong SLiu BGeng MFang Q(2023)Algorithms for maximizing monotone submodular function minus modular function under noiseJournal of Combinatorial Optimization10.1007/s10878-023-01026-545:4Online publication date: 19-Apr-2023
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Feldman MSvensson OZenklusen R(2022)A Framework for the Secretary Problem on the Intersection of MatroidsSIAM Journal on Computing10.1137/21M141182251:3(766-819)Online publication date: 14-Jun-2022
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