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.2022.113
URN: urn:nbn:de:0030-drops-157095
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Rubinstein, Aviad ; Zhao, Junyao

Budget-Smoothed Analysis for Submodular Maximization

LIPIcs-ITCS-2022-113.pdf (0.8 MB)


The greedy algorithm for monotone submodular function maximization subject to cardinality constraint is guaranteed to approximate the optimal solution to within a 1-1/e factor. Although it is well known that this guarantee is essentially tight in the worst case - for greedy and in fact any efficient algorithm, experiments show that greedy performs better in practice. We observe that for many applications in practice, the empirical distribution of the budgets (i.e., cardinality constraints) is supported on a wide range, and moreover, all the existing hardness results in theory break under a large perturbation of the budget.
To understand the effect of the budget from both algorithmic and hardness perspectives, we introduce a new notion of budget-smoothed analysis. We prove that greedy is optimal for every budget distribution, and we give a characterization for the worst-case submodular functions. Based on these results, we show that on the algorithmic side, under realistic budget distributions, greedy and related algorithms enjoy provably better approximation guarantees, that hold even for worst-case functions, and on the hardness side, there exist hard functions that are fairly robust to all the budget distributions.

BibTeX - Entry

  author =	{Rubinstein, Aviad and Zhao, Junyao},
  title =	{{Budget-Smoothed Analysis for Submodular Maximization}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{113:1--113:23},
  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-157095},
  doi =		{10.4230/LIPIcs.ITCS.2022.113},
  annote =	{Keywords: Submodular optimization, Beyond worst-case analysis, Greedy algorithms, Hardness of approximation}

Keywords: Submodular optimization, Beyond worst-case analysis, Greedy algorithms, Hardness of approximation
Collection: 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)
Issue Date: 2022
Date of publication: 25.01.2022

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