TY - GEN
T1 - A deterministic fully polynomial time approximation scheme for counting integer knapsack solutions made easy
AU - Halman, Nir
N1 - Funding Information: Partial support for this research was provided by the Recanati Fund of the Jerusalem School of Business Administration .
PY - 2016/9/1
Y1 - 2016/9/1
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.
AB - 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.
KW - Approximate counting
KW - Bounding constraints
KW - Dynamic programming
KW - Integer knapsack
KW - K-Approximating sets and functions.
UR - https://www.scopus.com/pages/publications/84990841383
U2 - 10.4230/LIPIcs.APPROX-RANDOM.2016.9
DO - 10.4230/LIPIcs.APPROX-RANDOM.2016.9
M3 - Conference contribution
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 19th International Workshop, APPROX 2016 and 20th International Workshop, RANDOM 2016
A2 - Jansen, Klaus
A2 - Mathieu, Claire
A2 - Rolim, Jose D. P.
A2 - Umans, Chris
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 19th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2016 and the 20th International Workshop on Randomization and Computation, RANDOM 2016
Y2 - 7 September 2016 through 9 September 2016
ER -