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DOI: 10.4230/LIPIcs.STACS.2008.1362

The Frobenius Problem in a Free Monoid

Authors: Jui-Yi Kao, Jeffrey Shallit, and Zhi Xu

Published in: LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)

Abstract

The classical Frobenius problem over ${mathbb N}$ is to compute the largest integer $g$ not representable as a non-negative integer linear combination of non-negative integers $x_1, x_2, ldots, x_k$, where $gcd(x_1, x_2, ldots, x_k) = 1$. In this paper we consider novel generalizations of the Frobenius problem to the noncommutative setting of a free monoid. Unlike the commutative case, where the bound on $g$ is quadratic, we are able to show exponential or subexponential behavior for several analogues of $g$, with the precise bound depending on the particular measure chosen.

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Jui-Yi Kao, Jeffrey Shallit, and Zhi Xu. The Frobenius Problem in a Free Monoid. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 421-432, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{kao_et_al:LIPIcs.STACS.2008.1362,
  author =	{Kao, Jui-Yi and Shallit, Jeffrey and Xu, Zhi},
  title =	{{The Frobenius Problem in a Free Monoid}},
  booktitle =	{25th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{421--432},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-06-4},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{1},
  editor =	{Albers, Susanne and Weil, Pascal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1362},
  URN =		{urn:nbn:de:0030-drops-13620},
  doi =		{10.4230/LIPIcs.STACS.2008.1362},
  annote =	{Keywords: Combinatorics on words, Frobenius problem, free monoid}
}

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The Frobenius Problem in a Free Monoid

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