A (1 − e−1 − ε)-approximation for the monotone submodular multiple knapsack problem

Yaron Fairstein; Ariel Kulik; Joseph Naor; Danny Raz; Hadas Shachnai

doi:10.4230/LIPIcs.ESA.2020.44

A (1 − e⁻¹ − ε)-approximation for the monotone submodular multiple knapsack problem

Yaron Fairstein, Ariel Kulik, Joseph Naor, Danny Raz, Hadas Shachnai

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

2 Scopus citations

Abstract

We study the problem of maximizing a monotone submodular function subject to a Multiple Knapsack constraint (SMKP) . The input is a set I of items, each associated with a non-negative weight, and a set of bins having arbitrary capacities. Also, we are given a submodular, monotone and non-negative function f over subsets of the items. The objective is to find a subset of items A ⊆ I and a packing of these items in the bins, such that f(A) is maximized. SMKP is a natural extension of both Multiple Knapsack and the problem of monotone submodular maximization subject to a knapsack constraint. Our main result is a nearly optimal polynomial time (1 − e⁻¹ − ε)-approximation algorithm for the problem, for any ε > 0. Our algorithm relies on a refined analysis of techniques for constrained submodular optimization combined with sophisticated application of tools used in the development of approximation schemes for packing problems.

Original language	English
Title of host publication	28th Annual European Symposium on Algorithms, ESA 2020
Editors	Fabrizio Grandoni, Grzegorz Herman, Peter Sanders
Publisher	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)	9783959771627
DOIs	https://doi.org/10.4230/LIPIcs.ESA.2020.44
State	Published - 1 Aug 2020
Externally published	Yes
Event	28th Annual European Symposium on Algorithms, ESA 2020 - Virtual, Pisa, Italy Duration: 7 Sep 2020 → 9 Sep 2020

Publication series

Name	Leibniz International Proceedings in Informatics, LIPIcs
Volume	173
ISSN (Print)	1868-8969

Conference

Conference	28th Annual European Symposium on Algorithms, ESA 2020
Country/Territory	Italy
City	Virtual, Pisa
Period	7/09/20 → 9/09/20

Keywords

Multiple Knapsack
Randomized Rounding
Sumodular Optimization

ASJC Scopus subject areas

Software

Access to Document

10.4230/LIPIcs.ESA.2020.44

Cite this

Fairstein, Y., Kulik, A., Naor, J., Raz, D., & Shachnai, H. (2020). A (1 − e⁻¹ − ε)-approximation for the monotone submodular multiple knapsack problem. In F. Grandoni, G. Herman, & P. Sanders (Eds.), 28th Annual European Symposium on Algorithms, ESA 2020 Article 44 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 173). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.ESA.2020.44

Fairstein, Yaron ; Kulik, Ariel ; Naor, Joseph et al. / A (1 − e⁻¹ − ε)-approximation for the monotone submodular multiple knapsack problem. 28th Annual European Symposium on Algorithms, ESA 2020. editor / Fabrizio Grandoni ; Grzegorz Herman ; Peter Sanders. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2020. (Leibniz International Proceedings in Informatics, LIPIcs).

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title = "A (1 − e−1 − ε)-approximation for the monotone submodular multiple knapsack problem",

abstract = "We study the problem of maximizing a monotone submodular function subject to a Multiple Knapsack constraint (SMKP) . The input is a set I of items, each associated with a non-negative weight, and a set of bins having arbitrary capacities. Also, we are given a submodular, monotone and non-negative function f over subsets of the items. The objective is to find a subset of items A ⊆ I and a packing of these items in the bins, such that f(A) is maximized. SMKP is a natural extension of both Multiple Knapsack and the problem of monotone submodular maximization subject to a knapsack constraint. Our main result is a nearly optimal polynomial time (1 − e−1 − ε)-approximation algorithm for the problem, for any ε > 0. Our algorithm relies on a refined analysis of techniques for constrained submodular optimization combined with sophisticated application of tools used in the development of approximation schemes for packing problems.",

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author = "Yaron Fairstein and Ariel Kulik and Joseph Naor and Danny Raz and Hadas Shachnai",

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Fairstein, Y, Kulik, A, Naor, J, Raz, D & Shachnai, H 2020, A (1 − e⁻¹ − ε)-approximation for the monotone submodular multiple knapsack problem. in F Grandoni, G Herman & P Sanders (eds), 28th Annual European Symposium on Algorithms, ESA 2020., 44, Leibniz International Proceedings in Informatics, LIPIcs, vol. 173, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 28th Annual European Symposium on Algorithms, ESA 2020, Virtual, Pisa, Italy, 7/09/20. https://doi.org/10.4230/LIPIcs.ESA.2020.44

A (1 − e⁻¹ − ε)-approximation for the monotone submodular multiple knapsack problem. / Fairstein, Yaron; Kulik, Ariel; Naor, Joseph et al.
28th Annual European Symposium on Algorithms, ESA 2020. ed. / Fabrizio Grandoni; Grzegorz Herman; Peter Sanders. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2020. 44 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 173).

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

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N2 - We study the problem of maximizing a monotone submodular function subject to a Multiple Knapsack constraint (SMKP) . The input is a set I of items, each associated with a non-negative weight, and a set of bins having arbitrary capacities. Also, we are given a submodular, monotone and non-negative function f over subsets of the items. The objective is to find a subset of items A ⊆ I and a packing of these items in the bins, such that f(A) is maximized. SMKP is a natural extension of both Multiple Knapsack and the problem of monotone submodular maximization subject to a knapsack constraint. Our main result is a nearly optimal polynomial time (1 − e−1 − ε)-approximation algorithm for the problem, for any ε > 0. Our algorithm relies on a refined analysis of techniques for constrained submodular optimization combined with sophisticated application of tools used in the development of approximation schemes for packing problems.

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Fairstein Y, Kulik A, Naor J, Raz D, Shachnai H. A (1 − e⁻¹ − ε)-approximation for the monotone submodular multiple knapsack problem. In Grandoni F, Herman G, Sanders P, editors, 28th Annual European Symposium on Algorithms, ESA 2020. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2020. 44. (Leibniz International Proceedings in Informatics, LIPIcs). doi: 10.4230/LIPIcs.ESA.2020.44