When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2017.50
URN: urn:nbn:de:0030-drops-80618
URL: https://drops.dagstuhl.de/opus/volltexte/2017/8061/
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### On Multidimensional and Monotone k-SUM

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### Abstract

The well-known k-SUM conjecture is that integer k-SUM requires time Omega(n^{\ceil{k/2}-o(1)}). Recent work has studied multidimensional k-SUM in F_p^d, where the best known algorithm takes time \tilde O(n^{\ceil{k/2}}). Bhattacharyya et al. [ICS 2011] proved a min(2^{\Omega(d)},n^{\Omega(k)}) lower bound for k-SUM in F_p^d under the Exponential Time Hypothesis. We give a more refined lower bound under the standard k-SUM conjecture: for sufficiently large p, k-SUM in F_p^d requires time Omega(n^{k/2-o(1)}) if k is even, and Omega(n^{\ceil{k/2}-2k(log k)/(log p)-o(1)}) if k is odd. For a special case of the multidimensional problem, bounded monotone d-dimensional 3SUM, Chan and Lewenstein [STOC 2015] gave a surprising \tilde O(n^{2-2/(d+13)}) algorithm using additive combinatorics. We show this algorithm is essentially optimal. To be more precise, bounded monotone d-dimensional 3SUM requires time Omega(n^{2-\frac{4}{d}-o(1)}) under the standard 3SUM conjecture, and time Omega(n^{2-\frac{2}{d}-o(1)}) under the so-called strong 3SUM conjecture. Thus, even though one might hope to further exploit the structural advantage of monotonicity, no substantial improvements beyond those obtained by Chan and Lewenstein are possible for bounded monotone d-dimensional 3SUM.

### BibTeX - Entry

@InProceedings{hsu_et_al:LIPIcs:2017:8061,
author =	{Chloe Ching-Yun Hsu and Chris Umans},
title =	{{On Multidimensional and Monotone k-SUM}},
booktitle =	{42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)},
pages =	{50:1--50:13},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-046-0},
ISSN =	{1868-8969},
year =	{2017},
volume =	{83},
editor =	{Kim G. Larsen and Hans L. Bodlaender and Jean-Francois Raskin},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},