Editing Open Problems:40
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# An undirected edge $(x,y)$ of the hypercube is ''violated'' if $|f(x) - f(y)| > 1$. Function $f$ is ''Lipschitz'' if no edges are violated.<br>'''Question:''' Suppose $v$ edges are violated. Give an upper bound on the number of node labels that have to be changed to make function $f$ Lipschitz in terms of $v$ and $d$.<br>'''Background:''' Nothing nontrivial is known for real labels. The conjecture is $O(v)$. | # An undirected edge $(x,y)$ of the hypercube is ''violated'' if $|f(x) - f(y)| > 1$. Function $f$ is ''Lipschitz'' if no edges are violated.<br>'''Question:''' Suppose $v$ edges are violated. Give an upper bound on the number of node labels that have to be changed to make function $f$ Lipschitz in terms of $v$ and $d$.<br>'''Background:''' Nothing nontrivial is known for real labels. The conjecture is $O(v)$. | ||
For integer labels, the best known bound is $2v \cdot {\rm ImageDiameter}(f)$, where ${\rm ImageDiameter}(f)=\max_x f(x) - \min_x f(x)$ {{cite|JhaR-11}}. | For integer labels, the best known bound is $2v \cdot {\rm ImageDiameter}(f)$, where ${\rm ImageDiameter}(f)=\max_x f(x) - \min_x f(x)$ {{cite|JhaR-11}}. | ||
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