On Submodular Prophet Inequalities and Correlation Gap

Prophet inequalities and secretary problems have been extensively studied in recent years due to their elegance, connections to online algorithms, stochastic optimization, and mechanism design problems in game theoretic settings. Rubinstein and Singla [37] developed a notion of combinatorial prophet inequalities in order to generalize the standard prophet inequality setting to combinatorial valuation functions such as submodular and subadditive functions. For non-negative submodular functions they demonstrated a constant factor prophet inequality for matroid constraints. Along the way they showed a variant of the correlation gap for non-negative submodular functions. In this paper we revisit their notion of correlation gap as well as the standard notion of correlation gap and prove much tighter and cleaner bounds. Via these bounds and other insights we obtain substantially improved constant factor combinatorial prophet inequalities for both monotone and non-monotone submodular functions over any constraint that admits an Online Contention Resolution Scheme. In addition to improved bounds we describe efficient polynomial-time algorithms that achieve these bounds.

[1]  Shuchi Chawla,et al.  Multi-parameter mechanism design and sequential posted pricing , 2010, BQGT.

[2]  Ruben Hoeksma,et al.  Recent developments in prophet inequalities , 2019, SECO.

[3]  Anupam Gupta,et al.  Random-Order Models , 2020, Beyond the Worst-Case Analysis of Algorithms.

[4]  Qiqi Yan,et al.  Mechanism design via correlation gap , 2010, SODA '11.

[5]  S. Matthew Weinberg,et al.  Matroid prophet inequalities , 2012, STOC '12.

[6]  Joseph Naor,et al.  A Unified Continuous Greedy Algorithm for Submodular Maximization , 2011, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[7]  Jason D. Hartline Bayesian Mechanism Design , 2013, Found. Trends Theor. Comput. Sci..

[8]  Saeed Alaei,et al.  Bayesian Combinatorial Auctions: Expanding Single Buyer Mechanisms to Many Buyers , 2011, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[9]  Brendan Lucier,et al.  An economic view of prophet inequalities , 2017, SECO.

[10]  Mohammad Taghi Hajiaghayi,et al.  Automated Online Mechanism Design and Prophet Inequalities , 2007, AAAI.

[11]  Niv Buchbinder,et al.  Submodular Functions Maximization Problems , 2018, Handbook of Approximation Algorithms and Metaheuristics.

[12]  Sahil Singla,et al.  (Near) Optimal Adaptivity Gaps for Stochastic Multi-Value Probing , 2019, APPROX-RANDOM.

[13]  Jan Vondrák,et al.  Maximizing a Monotone Submodular Function Subject to a Matroid Constraint , 2011, SIAM J. Comput..

[14]  Paul Dütting,et al.  Prophet Inequalities Made Easy: Stochastic Optimization by Pricing Nonstochastic Inputs , 2020, SIAM J. Comput..

[15]  Marek Adamczyk,et al.  Random Order Contention Resolution Schemes , 2018, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS).

[16]  Paul Dütting,et al.  An O(log log m) Prophet Inequality for Subadditive Combinatorial Auctions , 2020, 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS).

[17]  T. Hill,et al.  A Survey of Prophet Inequalities in Optimal Stopping Theory , 1992 .

[18]  Michal Feldman,et al.  Online Stochastic Max-Weight Matching: Prophet Inequality for Vertex and Edge Arrival Models , 2020, EC.

[19]  Mohammad Taghi Hajiaghayi,et al.  Prophet Secretary for Combinatorial Auctions and Matroids , 2017, SODA.

[20]  Aviad Rubinstein,et al.  Beyond matroids: secretary problem and prophet inequality with general constraints , 2016, STOC.

[21]  Amin Saberi,et al.  INFORMS doi 10.1287/xxxx.0000.0000 c○0000 INFORMS Price of Correlations in Stochastic Optimization , 2022 .

[22]  U. Krengel,et al.  Semiamarts and finite values , 1977 .

[23]  Anupam Gupta,et al.  A Stochastic Probing Problem with Applications , 2013, IPCO.

[24]  Ola Svensson,et al.  Online Contention Resolution Schemes , 2015, SODA.

[25]  Hamid Nazerzadeh,et al.  Maximizing Stochastic Monotone Submodular Functions , 2009, Manag. Sci..

[26]  Chandra Chekuri,et al.  Submodular function maximization via the multilinear relaxation and contention resolution schemes , 2011, STOC '11.

[27]  Joseph Naor,et al.  Submodular Maximization with Cardinality Constraints , 2014, SODA.

[28]  Jan Vondrák,et al.  Submodularity in Combinatorial Optimization , 2007 .

[29]  Michael Dinitz,et al.  Recent advances on the matroid secretary problem , 2013, SIGA.

[30]  Sepehr Assadi,et al.  Improved Truthful Mechanisms for Subadditive Combinatorial Auctions: Breaking the Logarithmic Barrier , 2020, SODA.

[31]  Maxim Sviridenko,et al.  Submodular Stochastic Probing on Matroids , 2013, Math. Oper. Res..

[32]  Sahil Singla,et al.  Combinatorial Prophet Inequalities , 2016, SODA.

[33]  Moran Feldman,et al.  The Submodular Secretary Problem Goes Linear , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[34]  Sahil Singla Combinatorial Optimization Under Uncertainty ( Probing and Stopping-Time Algorithms ) , 2017 .

[35]  Rama Chellappa,et al.  Entropy-Rate Clustering: Cluster Analysis via Maximizing a Submodular Function Subject to a Matroid Constraint , 2014, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[36]  Shuchi Chawla,et al.  Bayesian algorithmic mechanism design , 2014, SECO.