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29 March 2024
 
  » arxiv » 1505.8031

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On the Linear Extension Complexity of Regular n-gons
Arnaud Vandaele ; Nicolas Gillis ; François Glineur ;
Date 29 May 2015
AbstractIn this paper, we propose a new upper bound on the linear extension complexity of regular $n$-gons. It is based on the equivalence between the computation of (i) an extended formulation of size $r$ of a polytope $P$, and (ii) a rank-$r$ nonnegative factorization of a slack matrix of the polytope $P$. We provide explicit nonnegative factorizations for the slack matrix of any regular $n$-gons of size $2 lceil log_2(n) ceil - 1$ if $2^{k-1} < n leq 2^{k-1}+2^{k-2}$ for some integer $k$, and of size $2 lceil log_2(n) ceil$ if $2^{k-1}+2^{k-2} < n leq 2^{k}$. For $2^{k-1}+2^{k-2} < n leq 2^{k}$, our bound coincides with the best known upper bound of $2 leftlceil log_2(n) ight ceil$ by Fiorini, Rothvoss and Tiwary [Extended Formulations for Polygons, Discrete Comput. Geom. 48(3), pp. 658-668, 2012]. We conjecture that our upper bound is tight, which is suggested by numerical experiments for small $n$. Moreover, this improved upper bound allows us to close the gap with the best known lower bound for certain regular $n$-gons (namely, $9 leq n leq 14$ and $21 leq n leq 24$) hence allowing for the first time to determine their extension complexity.
Source arXiv, 1505.8031
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