A faster FPTAS for #knapsack

Paweł Gawrychowski, Liran Markin, Oren Weimann

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Given a set W = (w1, . . ., wn) 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 (u1, . . ., un) and we are allowed to take up to ui items of weight wi. We present a deterministic FPTAS for #Knapsack running in O(n2.5ε 1.5 log(nε 1) log(nε)) time. The previous best deterministic algorithm [FOCS 2011] runs in O(n3ε 1 log(nε 1)) time (see also [ESA 2014] for a logarithmic factor improvement). The previous best randomized algorithm [STOC 2003] runs in O(n2.5log(nε 1) + ε 2n2) time. Therefore, for the case of constant ε, we close the gap between the O(n2.5) randomized algorithm and the O(n3) deterministic algorithm. For the integer version with U = maxi (ui), we present a deterministic FPTAS running in O(n2.5ε 1.5 log(nε 1 log U) log(nε) log2 U) time. The previous best deterministic algorithm [TCS 2016] runs in O(n3ε 1 log(nε 1 log U) log2 U) time.

Original languageEnglish
Title of host publication45th International Colloquium on Automata, Languages, and Programming, ICALP 2018
EditorsChristos Kaklamanis, Daniel Marx, Ioannis Chatzigiannakis, Donald Sannella
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770767
DOIs
StatePublished - 1 Jul 2018
Event45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 - Prague, Czech Republic
Duration: 9 Jul 201813 Jul 2018

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume107
ISSN (Print)1868-8969

Conference

Conference45th International Colloquium on Automata, Languages, and Programming, ICALP 2018
Country/TerritoryCzech Republic
CityPrague
Period9/07/1813/07/18

Bibliographical note

Funding Information:
Supported in part by Israel Science Foundation grant 592/17 2 Supported in part by Israel Science Foundation grant 592/17

Publisher Copyright:
© Paweł Gawrychowski, Liran Markin, and Oren Weimann;.

Keywords

  • Approximate counting
  • Functions
  • K-approximating sets
  • Knapsack

ASJC Scopus subject areas

  • Software

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