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

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 alg...

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Bibliographic Details
Published in:Theoretical computer science Vol. 645; pp. 41 - 47
Main Author: Halman, Nir
Format: Journal Article
Language:English
Published: Elsevier B.V 13.09.2016
Subjects:
Algorithms
Approximate counting
Approximation
Binding
Binding constraints
Dynamic programming
Estimates
Integer knapsack
Integers
K-approximating sets and functions
Mathematical analysis
Mathematical models
Polynomials
Integer knapsack
Dynamic programming
Binding constraints
K-approximating sets and functions
Approximate counting
ISSN:0304-3975, 1879-2294
Online Access:Get full text
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Summary: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(n3log2⁡Uϵlog⁡nlog⁡Uϵ) time. This is an improvement of n2 and 1/ϵ (up to log terms) over the best known deterministic algorithm by Gopalan et al. (2011) [5]. 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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ISSN:0304-3975
1879-2294
DOI:10.1016/j.tcs.2016.06.015