License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2016.13
URN: urn:nbn:de:0030-drops-57143
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/5714/
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Austrin, Per ; Kaski, Petteri ; Koivisto, Mikko ; Nederlof, Jesper

Dense Subset Sum May Be the Hardest

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Abstract

The SUBSET SUM problem asks whether a given set of n positive integers contains a subset of elements that sum up to a given target t. It is an outstanding open question whether the O^*(2^{n/2})-time algorithm for SUBSET SUM by Horowitz and Sahni [J. ACM 1974] can be beaten in the worst-case setting by a "truly faster", O^*(2^{(0.5-delta)*n})-time algorithm, with some constant delta > 0. Continuing an earlier work [STACS 2015], we study SUBSET SUM parameterized by the maximum bin size beta, defined as the largest number of subsets of the n input integers that yield the same sum. For every epsilon > 0 we give a truly faster algorithm for instances with beta <= 2^{(0.5-epsilon)*n}, as well as instances with beta >= 2^{0.661n}. Consequently, we also obtain a characterization in terms of the popular density parameter n/log_2(t): if all instances of density at least 1.003 admit a truly faster algorithm, then so does every instance. This goes against the current intuition that instances of density 1 are the hardest, and therefore is a step toward answering the open question in the affirmative. Our results stem from a novel combinatorial analysis of mixings of earlier algorithms for SUBSET SUM and a study of an extremal question in additive combinatorics connected to the problem of Uniquely Decodable Code Pairs in information theory.

BibTeX - Entry

@InProceedings{austrin_et_al:LIPIcs:2016:5714,
  author =	{Per Austrin and Petteri Kaski and Mikko Koivisto and Jesper Nederlof},
  title =	{{Dense Subset Sum May Be the Hardest}},
  booktitle =	{33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)},
  pages =	{13:1--13:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-001-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{47},
  editor =	{Nicolas Ollinger and Heribert Vollmer},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/5714},
  URN =		{urn:nbn:de:0030-drops-57143},
  doi =		{10.4230/LIPIcs.STACS.2016.13},
  annote =	{Keywords: subset sum, additive combinatorics, exponential-time algorithm, homo-morphic hashing, littlewood–offord problem}
}

Keywords: subset sum, additive combinatorics, exponential-time algorithm, homo-morphic hashing, littlewood–offord problem
Collection: 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)
Issue Date: 2016
Date of publication: 16.02.2016

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