### Approximating the bandwidth for asteroidal triple-free graphs [chapter]

T. Kloks, D. Kratsch, H. Müller
<span title="">1995</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
We show that there is an O(n 3 ) algorithm to approximate the bandwidth of an AT-free graph with worst case performance ratio 2. Alternatively, at the cost of the approximation factor, we can also obtain an O(e + nlog n) algorithm to approximate the bandwidth of an AT-free graph within a factor 4 and an O(n + e) algorithm with a factor 6. For the special cases of permutation graphs and trapezoid graphs we obtain O(nlog 2 n) algorithms with worst case performance ratio 2. For cocomparability
more &raquo; ... hs we obtain an O(n + e) algorithm with worst case performance ratio 3. Finally, we show that there is an O(n 2 log 2 n) algorithm to compute the exact bandwidth of chain graphs. Theorem 2.4 The problem'Given a cobipartite graph G = (V; E) and a positive integer k, decide whether bw(G) k', is NP-complete. Proof. Let G be a cobipartite graph. Then (G) 2 implies that G is claw-free. Hence G is claw-free and AT-free and this implies bw(G) = tw(G) by Theorem 3.12. Finally the problem TREEWIDTH is known to be NP-complete on cobipartite graphs  implying that BANDWIDTH on cobipartite graphs is NP-complete. 2 3 Clearly this implies that BANDWIDTH remains NP-complete when restricted to cocomparability graphs or to AT-free graphs. Most of our approximation algorithms are based on the following theorem. Here
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