License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2022.120
URN: urn:nbn:de:0030-drops-157169
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Wang, Weina ; Gupta, Anupam ; Williams, Jalani K.

Probing to Minimize

LIPIcs-ITCS-2022-120.pdf (1 MB)


We develop approximation algorithms for set-selection problems with deterministic constraints, but random objective values, i.e., stochastic probing problems. When the goal is to maximize the objective, approximation algorithms for probing problems are well-studied. On the other hand, few techniques are known for minimizing the objective, especially in the adaptive setting, where information about the random objective is revealed during the set-selection process and allowed to influence it. For minimization problems in particular, incorporating adaptivity can have a considerable effect on performance. In this work, we seek approximation algorithms that compare well to the optimal adaptive policy.
We develop new techniques for adaptive minimization, applying them to a few problems of interest. The core technique we develop here is an approximate reduction from an adaptive expectation minimization problem to a set of adaptive probability minimization problems which we call threshold problems. By providing near-optimal solutions to these threshold problems, we obtain bicriteria adaptive policies.
We apply this method to obtain an adaptive approximation algorithm for the Min-Element problem, where the goal is to adaptively pick random variables to minimize the expected minimum value seen among them, subject to a knapsack constraint. This partially resolves an open problem raised in [Goel et al., 2010]. We further consider three extensions on the Min-Element problem, where our objective is the sum of the smallest k element-weights, or the weight of the min-weight basis of a given matroid, or where the constraint is not given by a knapsack but by a matroid constraint. For all three of the variations we explore, we develop adaptive approximation algorithms for their corresponding threshold problems, and prove their near-optimality via coupling arguments.

BibTeX - Entry

  author =	{Wang, Weina and Gupta, Anupam and Williams, Jalani K.},
  title =	{{Probing to Minimize}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{120:1--120:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-157169},
  doi =		{10.4230/LIPIcs.ITCS.2022.120},
  annote =	{Keywords: approximation algorithms, stochastic probing, minimization}

Keywords: approximation algorithms, stochastic probing, minimization
Collection: 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)
Issue Date: 2022
Date of publication: 25.01.2022

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