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DOI: 10.4230/LIPIcs.ICALP.2018.64
A Faster FPTAS for #Knapsack

Authors: Pawel Gawrychowski, Liran Markin, and Oren Weimann

Published in: LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

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.
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)

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@InProceedings{gawrychowski_et_al:LIPIcs.ICALP.2018.64,
  author =	{Gawrychowski, Pawel and Markin, Liran and Weimann, Oren},
  title =	{{A Faster FPTAS for #Knapsack}},
  booktitle =	{45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
  pages =	{64:1--64:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.64},
  URN =		{urn:nbn:de:0030-drops-90687},
  doi =		{10.4230/LIPIcs.ICALP.2018.64},
  annote =	{Keywords: knapsack, approximate counting, K-approximating sets and functions}
}
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