Document

# A Faster FPTAS for #Knapsack

## File

LIPIcs.ICALP.2018.64.pdf
• Filesize: 484 kB
• 13 pages

## Cite As

Pawel Gawrychowski, Liran Markin, and Oren Weimann. A Faster FPTAS for #Knapsack. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 64:1-64:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ICALP.2018.64

## Abstract

Given a set W = {w_1,..., w_n} of non-negative integer weights and an integer C, the #Knapsack problem asks to count the number of distinct subsets of W whose total weight is at most C. In the more general integer version of the problem, the subsets are multisets. That is, we are also given a set {u_1,..., u_n} and we are allowed to take up to u_i items of weight w_i. We present a deterministic FPTAS for #Knapsack running in O(n^{2.5}epsilon^{-1.5}log(n epsilon^{-1})log (n epsilon)) time. The previous best deterministic algorithm [FOCS 2011] runs in O(n^3 epsilon^{-1} log(n epsilon^{-1})) time (see also [ESA 2014] for a logarithmic factor improvement). The previous best randomized algorithm [STOC 2003] runs in O(n^{2.5} sqrt{log (n epsilon^{-1})} + epsilon^{-2} n^2) time. Therefore, for the case of constant epsilon, we close the gap between the O~(n^{2.5}) randomized algorithm and the O~(n^3) deterministic algorithm. For the integer version with U = max_i {u_i}, we present a deterministic FPTAS running in O(n^{2.5}epsilon^{-1.5}log(n epsilon^{-1} log U)log (n epsilon) log^2 U) time. The previous best deterministic algorithm [TCS 2016] runs in O(n^3 epsilon^{-1}log(n epsilon^{-1} log U) log^2 U) time.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
##### Keywords
• knapsack
• approximate counting
• K-approximating sets and functions

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Mohsen Bayati, David Gamarnik, Dimitriy Katz, Chandra Nair, and Prasad Tetali. Simple deterministic approximation algorithms for counting matchings. In STOC, pages 122-127, 2007.
2. Ilias Diakonikolas, Parikshit Gopalan, Ragesh Jaiswal, Rocco A. Servedio, and Emanuele Viola. Bounded independence fools halfspaces. In FOCS, pages 171-180, 2009.
3. Martin Dyer. Approximate counting by dynamic programming. In STOC, pages 693-699, 2003.
4. Martin Dyer, Alan Frieze, Ravi Kannan, Ajai Kapoor, Ljubomir Perkovic, and Umesh Vazirani. A mildly exponential time algorithm for approximating the number of solutions to a multidimensional knapsack problem. Combinatorics, Probability and Computing, 2:271-284, 1993.
5. Parikshit Gopalan, Adam Klivans, and Raghu Meka. Polynomial-time approximation schemes for knapsack and related counting problems using branching programs. arXiv 1008.3187, 2010.
6. Parikshit Gopalan, Adam Klivans, Raghu Meka, Daniel Štefankovic, Santosh Vempala, and Eric Vigoda. An FPTAS for #Knapsack and related counting problems. In FOCS, pages 817-826, 2011.
7. Nir Halman. A deterministic fully polynomial time approximation scheme for counting integer knapsack solutions made easy. Theoretical Computer Science, 645:41-47, 2016.
8. Nir Halman, Diego Klabjan, Mohamed Mostagir, Jim Orlin, and David Simchi-Levi. A fully polynomial-time approximation scheme for single-item stochastic inventory control with discrete demand. Mathematics of Operations Research, 34(3):674-685, 2009.
9. Raghu Meka and David Zuckerman. Pseudorandom generators for polynomial threshold functions. In STOC, pages 427-436, 2010.
10. Ben Morris and Alistair Sinclair. Random walks on truncated cubes and sampling 0-1 knapsack solutions. SIAM journal on computing, 34(1):195-226, 2004. Preliminary version in FOCS 1999.
11. Yuval Rabani and Amir Shpilka. Explicit construction of a small epsilon-net for linear threshold functions. In STOC, pages 649-658, 2009.
12. Romeo Rizzi and Alexandru I. Tomescu. Faster FPTASes for counting and random generation of knapsack solutions. In ESA, pages 762-773, 2014.
13. Daniel Štefankovič, Santosh Vempala, and Eric Vigoda. A deterministic polynomial-time approximation scheme for counting knapsack solutions. SIAM Journal on Computing, 41(2):356-366, 2012.
14. Dror Weitz. Counting independent sets up to the tree threshold. In STOC, pages 140-149, 2006.
X

Feedback for Dagstuhl Publishing