Maximizing Sums of Non-Monotone Submodular and Linear Functions: Understanding the Unconstrained Case

Kobi Bodek; Moran Feldman

doi:10.4230/LIPIcs.ESA.2022.23

Maximizing Sums of Non-Monotone Submodular and Linear Functions: Understanding the Unconstrained Case

Kobi Bodek, Moran Feldman

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

Motivated by practical applications, recent works have considered maximization of sums of a submodular function g and a linear function ℓ. Almost all such works, to date, studied only the special case of this problem in which g is also guaranteed to be monotone. Therefore, in this paper we systematically study the simplest version of this problem in which g is allowed to be non-monotone, namely the unconstrained variant, which we term Regularized Unconstrained Submodular Maximization (RegularizedUSM). Our main algorithmic result is the first non-trivial guarantee for general RegularizedUSM. For the special case of RegularizedUSM in which the linear function ℓ is non-positive, we prove two inapproximability results, showing that the algorithmic result implied for this case by previous works is not far from optimal. Finally, we reanalyze the known Double Greedy algorithm to obtain improved guarantees for the special case of RegularizedUSM in which the linear function ℓ is non-negative; and we complement these guarantees by showing that it is not possible to obtain (1/2, 1)-approximation for this case (despite intuitive arguments suggesting that this approximation guarantee is natural).

Original language	English
Title of host publication	30th Annual European Symposium on Algorithms, ESA 2022
Editors	Shiri Chechik, Gonzalo Navarro, Eva Rotenberg, Grzegorz Herman
Publisher	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages	23:1-23:17
Number of pages	17
ISBN (Electronic)	9783959772471
DOIs	https://doi.org/10.4230/LIPIcs.ESA.2022.23
State	Published - 1 Sep 2022
Externally published	Yes
Event	30th Annual European Symposium on Algorithms, ESA 2022 - Berlin/Potsdam, Germany Duration: 5 Sep 2022 → 9 Sep 2022

Publication series

Name	Leibniz International Proceedings in Informatics, LIPIcs
Volume	244
ISSN (Print)	1868-8969

Conference

Conference	30th Annual European Symposium on Algorithms, ESA 2022
Country/Territory	Germany
City	Berlin/Potsdam
Period	5/09/22 → 9/09/22

Bibliographical note

Publisher Copyright:
© 2022 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.

Keywords

Unconstrained submodular maximization
double greedy
inapproximability
non-oblivious local search
regularization

Access to Document

10.4230/LIPIcs.ESA.2022.23

Cite this

Bodek, K., & Feldman, M. (2022). Maximizing Sums of Non-Monotone Submodular and Linear Functions: Understanding the Unconstrained Case. In S. Chechik, G. Navarro, E. Rotenberg, & G. Herman (Eds.), 30th Annual European Symposium on Algorithms, ESA 2022 (pp. 23:1-23:17). (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 244). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.ESA.2022.23

Bodek, Kobi ; Feldman, Moran. / Maximizing Sums of Non-Monotone Submodular and Linear Functions : Understanding the Unconstrained Case. 30th Annual European Symposium on Algorithms, ESA 2022. editor / Shiri Chechik ; Gonzalo Navarro ; Eva Rotenberg ; Grzegorz Herman. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2022. pp. 23:1-23:17 (Leibniz International Proceedings in Informatics, LIPIcs).

@inproceedings{796b796c836a4f0996c210f9ecee1015,

title = "Maximizing Sums of Non-Monotone Submodular and Linear Functions: Understanding the Unconstrained Case",

abstract = "Motivated by practical applications, recent works have considered maximization of sums of a submodular function g and a linear function ℓ. Almost all such works, to date, studied only the special case of this problem in which g is also guaranteed to be monotone. Therefore, in this paper we systematically study the simplest version of this problem in which g is allowed to be non-monotone, namely the unconstrained variant, which we term Regularized Unconstrained Submodular Maximization (RegularizedUSM). Our main algorithmic result is the first non-trivial guarantee for general RegularizedUSM. For the special case of RegularizedUSM in which the linear function ℓ is non-positive, we prove two inapproximability results, showing that the algorithmic result implied for this case by previous works is not far from optimal. Finally, we reanalyze the known Double Greedy algorithm to obtain improved guarantees for the special case of RegularizedUSM in which the linear function ℓ is non-negative; and we complement these guarantees by showing that it is not possible to obtain (1/2, 1)-approximation for this case (despite intuitive arguments suggesting that this approximation guarantee is natural).",

keywords = "Unconstrained submodular maximization, double greedy, inapproximability, non-oblivious local search, regularization",

author = "Kobi Bodek and Moran Feldman",

note = "Publisher Copyright: {\textcopyright} 2022 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.; 30th Annual European Symposium on Algorithms, ESA 2022 ; Conference date: 05-09-2022 Through 09-09-2022",

year = "2022",

month = sep,

day = "1",

doi = "10.4230/LIPIcs.ESA.2022.23",

language = "אנגלית",

series = "Leibniz International Proceedings in Informatics, LIPIcs",

publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",

pages = "23:1--23:17",

editor = "Shiri Chechik and Gonzalo Navarro and Eva Rotenberg and Grzegorz Herman",

booktitle = "30th Annual European Symposium on Algorithms, ESA 2022",

address = "גרמניה",

}

Bodek, K & Feldman, M 2022, Maximizing Sums of Non-Monotone Submodular and Linear Functions: Understanding the Unconstrained Case. in S Chechik, G Navarro, E Rotenberg & G Herman (eds), 30th Annual European Symposium on Algorithms, ESA 2022. Leibniz International Proceedings in Informatics, LIPIcs, vol. 244, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, pp. 23:1-23:17, 30th Annual European Symposium on Algorithms, ESA 2022, Berlin/Potsdam, Germany, 5/09/22. https://doi.org/10.4230/LIPIcs.ESA.2022.23

Maximizing Sums of Non-Monotone Submodular and Linear Functions: Understanding the Unconstrained Case. / Bodek, Kobi; Feldman, Moran.
30th Annual European Symposium on Algorithms, ESA 2022. ed. / Shiri Chechik; Gonzalo Navarro; Eva Rotenberg; Grzegorz Herman. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2022. p. 23:1-23:17 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 244).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

TY - GEN

T1 - Maximizing Sums of Non-Monotone Submodular and Linear Functions

T2 - 30th Annual European Symposium on Algorithms, ESA 2022

AU - Bodek, Kobi

AU - Feldman, Moran

PY - 2022/9/1

Y1 - 2022/9/1

N2 - Motivated by practical applications, recent works have considered maximization of sums of a submodular function g and a linear function ℓ. Almost all such works, to date, studied only the special case of this problem in which g is also guaranteed to be monotone. Therefore, in this paper we systematically study the simplest version of this problem in which g is allowed to be non-monotone, namely the unconstrained variant, which we term Regularized Unconstrained Submodular Maximization (RegularizedUSM). Our main algorithmic result is the first non-trivial guarantee for general RegularizedUSM. For the special case of RegularizedUSM in which the linear function ℓ is non-positive, we prove two inapproximability results, showing that the algorithmic result implied for this case by previous works is not far from optimal. Finally, we reanalyze the known Double Greedy algorithm to obtain improved guarantees for the special case of RegularizedUSM in which the linear function ℓ is non-negative; and we complement these guarantees by showing that it is not possible to obtain (1/2, 1)-approximation for this case (despite intuitive arguments suggesting that this approximation guarantee is natural).

AB - Motivated by practical applications, recent works have considered maximization of sums of a submodular function g and a linear function ℓ. Almost all such works, to date, studied only the special case of this problem in which g is also guaranteed to be monotone. Therefore, in this paper we systematically study the simplest version of this problem in which g is allowed to be non-monotone, namely the unconstrained variant, which we term Regularized Unconstrained Submodular Maximization (RegularizedUSM). Our main algorithmic result is the first non-trivial guarantee for general RegularizedUSM. For the special case of RegularizedUSM in which the linear function ℓ is non-positive, we prove two inapproximability results, showing that the algorithmic result implied for this case by previous works is not far from optimal. Finally, we reanalyze the known Double Greedy algorithm to obtain improved guarantees for the special case of RegularizedUSM in which the linear function ℓ is non-negative; and we complement these guarantees by showing that it is not possible to obtain (1/2, 1)-approximation for this case (despite intuitive arguments suggesting that this approximation guarantee is natural).

KW - Unconstrained submodular maximization

KW - double greedy

KW - inapproximability

KW - non-oblivious local search

KW - regularization

UR - http://www.scopus.com/inward/record.url?scp=85137543583&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ESA.2022.23

DO - 10.4230/LIPIcs.ESA.2022.23

M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.conference???

AN - SCOPUS:85137543583

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 23:1-23:17

BT - 30th Annual European Symposium on Algorithms, ESA 2022

A2 - Chechik, Shiri

A2 - Navarro, Gonzalo

A2 - Rotenberg, Eva

A2 - Herman, Grzegorz

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

Y2 - 5 September 2022 through 9 September 2022

ER -

Bodek K, Feldman M. Maximizing Sums of Non-Monotone Submodular and Linear Functions: Understanding the Unconstrained Case. In Chechik S, Navarro G, Rotenberg E, Herman G, editors, 30th Annual European Symposium on Algorithms, ESA 2022. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2022. p. 23:1-23:17. (Leibniz International Proceedings in Informatics, LIPIcs). doi: 10.4230/LIPIcs.ESA.2022.23

Maximizing Sums of Non-Monotone Submodular and Linear Functions: Understanding the Unconstrained Case

Abstract

Publication series

Conference

Bibliographical note

Keywords

Access to Document

Other files and links

Fingerprint

Cite this