PapersWe consider the NP-complete optimization problem Bandwidth. Anupam Gupta gave an $O(\log^{2.5} n)$ approximation algorithm for trees, and showed that his algorithm has an approximation ratio of $O(\log n)$ on caterpillars, trees composed of a central path and paths emanating from it. We show that the same approximation ratio is obtained on trees composed of a central path and caterpillars emanating from it.
Our result relies on the following lemma.
Definition. A sequence $a_1,\ldots,a_n$ has thickness $\Theta$ if the sum of any $d$ consecutive elements is at most $d\Theta$, for $1 \leq d \leq n$.
Lemma. If a sequence has thickness $\Theta$, then the sequence obtained by ordering the elements in non-decreasing order also has thickness $\Theta$.
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