A deterministic fully polynomial time approximation scheme for counting integer knapsack solutions made easy

Nir Halman

doi:10.4230/LIPIcs.APPROX-RANDOM.2016.9

A deterministic fully polynomial time approximation scheme for counting integer knapsack solutions made easy

Nir Halman

Hebrew University of Jerusalem

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

Abstract

Given n elements with nonnegative integer weights w = (w1, . . . ,wn), an integer capacity C and positive integer ranges u = (u1, . . . , un), we consider the counting version of the classic integer knapsack problem: find the number of distinct multisets whose weights add up to at most C. We give a deterministic algorithm that estimates the number of solutions to within relative error ϵ in time polynomial in n, log U and 1/ϵ, where U = maxi ui. More precisely, our algorithm runs in O( n³ log² U/ϵ log n log U ϵ ) time. This is an improvement of n² and 1/² (up to log terms) over the best known deterministic algorithm by Gopalan et al. [FOCS, (2011), pp. 817-826]. Our algorithm is relatively simple, and its analysis is rather elementary. Our results are achieved by means of a careful formulation of the problem as a dynamic program, using the notion of binding constraints.

Original language	English
Title of host publication	Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 19th International Workshop, APPROX 2016 and 20th International Workshop, RANDOM 2016
Editors	Klaus Jansen, Claire Mathieu, Jose D. P. Rolim, Chris Umans
Publisher	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)	9783959770187
DOIs	https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2016.9
State	Published - 1 Sep 2016
Externally published	Yes
Event	19th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2016 and the 20th International Workshop on Randomization and Computation, RANDOM 2016 - Paris, France Duration: 7 Sep 2016 → 9 Sep 2016

Publication series

Name	Leibniz International Proceedings in Informatics, LIPIcs
Volume	60
ISSN (Print)	1868-8969

Conference

Conference	19th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2016 and the 20th International Workshop on Randomization and Computation, RANDOM 2016
Country/Territory	France
City	Paris
Period	7/09/16 → 9/09/16

Bibliographical note

Funding Information:
Partial support for this research was provided by the Recanati Fund of the Jerusalem School of Business Administration .

Keywords

Approximate counting
Bounding constraints
Dynamic programming
Integer knapsack
K-Approximating sets and functions.

Access to Document

10.4230/LIPIcs.APPROX-RANDOM.2016.9

Cite this

Halman, N. (2016). A deterministic fully polynomial time approximation scheme for counting integer knapsack solutions made easy. In K. Jansen, C. Mathieu, J. D. P. Rolim, & C. Umans (Eds.), Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 19th International Workshop, APPROX 2016 and 20th International Workshop, RANDOM 2016 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 60). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2016.9

Halman, Nir. / A deterministic fully polynomial time approximation scheme for counting integer knapsack solutions made easy. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 19th International Workshop, APPROX 2016 and 20th International Workshop, RANDOM 2016. editor / Klaus Jansen ; Claire Mathieu ; Jose D. P. Rolim ; Chris Umans. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2016. (Leibniz International Proceedings in Informatics, LIPIcs).

@inproceedings{588be929b81f4705a3e0ad100dcc3882,

title = "A deterministic fully polynomial time approximation scheme for counting integer knapsack solutions made easy",

abstract = "Given n elements with nonnegative integer weights w = (w1, . . . ,wn), an integer capacity C and positive integer ranges u = (u1, . . . , un), we consider the counting version of the classic integer knapsack problem: find the number of distinct multisets whose weights add up to at most C. We give a deterministic algorithm that estimates the number of solutions to within relative error ϵ in time polynomial in n, log U and 1/ϵ, where U = maxi ui. More precisely, our algorithm runs in O( n3 log2 U/ϵ log n log U ϵ ) time. This is an improvement of n2 and 1/2 (up to log terms) over the best known deterministic algorithm by Gopalan et al. [FOCS, (2011), pp. 817-826]. Our algorithm is relatively simple, and its analysis is rather elementary. Our results are achieved by means of a careful formulation of the problem as a dynamic program, using the notion of binding constraints.",

keywords = "Approximate counting, Bounding constraints, Dynamic programming, Integer knapsack, K-Approximating sets and functions.",

author = "Nir Halman",

note = "Funding Information: Partial support for this research was provided by the Recanati Fund of the Jerusalem School of Business Administration . ; 19th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2016 and the 20th International Workshop on Randomization and Computation, RANDOM 2016 ; Conference date: 07-09-2016 Through 09-09-2016",

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Halman, N 2016, A deterministic fully polynomial time approximation scheme for counting integer knapsack solutions made easy. in K Jansen, C Mathieu, JDP Rolim & C Umans (eds), Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 19th International Workshop, APPROX 2016 and 20th International Workshop, RANDOM 2016. Leibniz International Proceedings in Informatics, LIPIcs, vol. 60, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 19th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2016 and the 20th International Workshop on Randomization and Computation, RANDOM 2016, Paris, France, 7/09/16. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2016.9

A deterministic fully polynomial time approximation scheme for counting integer knapsack solutions made easy. / Halman, Nir.
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 19th International Workshop, APPROX 2016 and 20th International Workshop, RANDOM 2016. ed. / Klaus Jansen; Claire Mathieu; Jose D. P. Rolim; Chris Umans. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2016. (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 60).

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

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N2 - Given n elements with nonnegative integer weights w = (w1, . . . ,wn), an integer capacity C and positive integer ranges u = (u1, . . . , un), we consider the counting version of the classic integer knapsack problem: find the number of distinct multisets whose weights add up to at most C. We give a deterministic algorithm that estimates the number of solutions to within relative error ϵ in time polynomial in n, log U and 1/ϵ, where U = maxi ui. More precisely, our algorithm runs in O( n3 log2 U/ϵ log n log U ϵ ) time. This is an improvement of n2 and 1/2 (up to log terms) over the best known deterministic algorithm by Gopalan et al. [FOCS, (2011), pp. 817-826]. Our algorithm is relatively simple, and its analysis is rather elementary. Our results are achieved by means of a careful formulation of the problem as a dynamic program, using the notion of binding constraints.

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Halman N. A deterministic fully polynomial time approximation scheme for counting integer knapsack solutions made easy. In Jansen K, Mathieu C, Rolim JDP, Umans C, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 19th International Workshop, APPROX 2016 and 20th International Workshop, RANDOM 2016. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2016. (Leibniz International Proceedings in Informatics, LIPIcs). doi: 10.4230/LIPIcs.APPROX-RANDOM.2016.9

A deterministic fully polynomial time approximation scheme for counting integer knapsack solutions made easy

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Keywords

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