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(Near) Optimal Adaptivity Gaps for Stochastic Multi-Value Probing

Authors Domagoj Bradac , Sahil Singla , Goran Zuzic

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Author Details

Domagoj Bradac
  • Department of Mathematics, Faculty of Science, University of Zagreb, Croatia
Sahil Singla
  • Department of Computer Science, Princeton University, NJ, USA
Goran Zuzic
  • Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA

Funding

  • Bradac, Domagoj: The Author thanks the Computer Science Department at Carnegie Mellon University for their support; part of his work was done with visiting in Summer 2018.
  • Singla, Sahil: Supported in part by NSF awards CCF-1319811, CCF-1536002, and CCF-1617790 and in part by the Schmidt foundation.
  • Zuzic, Goran: Supported in part by NSF grants CCF-1527110, CCF-1618280, CCF-1814603, CCF-1910588, NSF CAREER award CCF-1750808 and a Sloan Research Fellowship.

Acknowledgements

We would like to thank Anupam Gupta for helpful discussions.

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Domagoj Bradac, Sahil Singla, and Goran Zuzic. (Near) Optimal Adaptivity Gaps for Stochastic Multi-Value Probing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 49:1-49:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.49

Abstract

Consider a kidney-exchange application where we want to find a max-matching in a random graph. To find whether an edge e exists, we need to perform an expensive test, in which case the edge e appears independently with a known probability p_e. Given a budget on the total cost of the tests, our goal is to find a testing strategy that maximizes the expected maximum matching size.
The above application is an example of the stochastic probing problem. In general the optimal stochastic probing strategy is difficult to find because it is adaptive - decides on the next edge to probe based on the outcomes of the probed edges. An alternate approach is to show the adaptivity gap is small, i.e., the best non-adaptive strategy always has a value close to the best adaptive strategy. This allows us to focus on designing non-adaptive strategies that are much simpler. Previous works, however, have focused on Bernoulli random variables that can only capture whether an edge appears or not. In this work we introduce a multi-value stochastic probing problem, which can also model situations where the weight of an edge has a probability distribution over multiple values.
Our main technical contribution is to obtain (near) optimal bounds for the (worst-case) adaptivity gaps for multi-value stochastic probing over prefix-closed constraints. For a monotone submodular function, we show the adaptivity gap is at most 2 and provide a matching lower bound. For a weighted rank function of a k-extendible system (a generalization of intersection of k matroids), we show the adaptivity gap is between O(k log k) and k. None of these results were known even in the Bernoulli case where both our upper and lower bounds also apply, thereby resolving an open question of Gupta et al. [Gupta et al., 2017].

Subject Classification

ACM Subject Classification
  • Theory of computation → Stochastic control and optimization
  • Theory of computation → Submodular optimization and polymatroids
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Design and analysis of algorithms
Keywords
  • stochastic programming
  • adaptivity gaps
  • stochastic multi-value probing
  • submodular functions
  • k-extendible systems
  • adaptive strategy
  • matroid intersection

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