When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ISAAC.2022.52
URN: urn:nbn:de:0030-drops-173372
URL: https://drops.dagstuhl.de/opus/volltexte/2022/17337/
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### Polynomial Threshold Functions for Decision Lists

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### Abstract

For S ⊆ {0,1}ⁿ a Boolean function f : S → {-1,1} is a polynomial threshold function (PTF) of degree d and weight W if there is a polynomial p with integer coefficients of degree d and with sum of absolute coefficients W such that f(x) = sign p(x) for all x ∈ S. We study a representation of decision lists as PTFs over Boolean cubes {0,1}ⁿ and over Hamming balls {0,1}ⁿ_{≤ k}.
As our first result, we show that for all d = O((n/(log n))^{1/3}) any decision list over {0,1}ⁿ can be represented by a PTF of degree d and weight 2^O(n/d²). This improves the result by Klivans and Servedio [Adam R. Klivans and Rocco A. Servedio, 2006] by a log² d factor in the exponent of the weight. Our bound is tight for all d = O((n/(log n))^{1/3}) due to the matching lower bound by Beigel [Richard Beigel, 1994].
For decision lists over a Hamming ball {0,1}ⁿ_{≤ k} we show that the upper bound on weight above can be drastically improved to n^O(√k) for d = Θ(√k). We also show that similar improvement is not possible for smaller degrees by proving the lower bound W = 2^Ω(n/d²) for all d = O(√k).

### BibTeX - Entry

```@InProceedings{podolskii_et_al:LIPIcs.ISAAC.2022.52,
author =	{Podolskii, Vladimir and Proskurin, Nikolay V.},
title =	{{Polynomial Threshold Functions for Decision Lists}},
booktitle =	{33rd International Symposium on Algorithms and Computation (ISAAC 2022)},
pages =	{52:1--52:12},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-258-7},
ISSN =	{1868-8969},
year =	{2022},
volume =	{248},
editor =	{Bae, Sang Won and Park, Heejin},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},