Abstract
We study the classical Bandwidth problem from the viewpoint of parametrised algorithms. Given a graph G = ( V , E ) and a positive integer k , the Bandwidth problem asks whether there exists a bijective function β : { 1 , … , ∣ V ∣ } → V such that for every edge u v ∈ E , ∣ β − 1 ( u ) − β − 1 ( v ) ∣ ≤ k . It is known that under standard complexity assumptions, no algorithm for Bandwidth with running time of the form f ( k ) n O ( 1 ) exists, even when the input is restricted to trees. We initiate the search for classes of graphs where such algorithms do exist. We present an algorithm with running time n ⋅ 2 O ( k log k ) for Bandwidth on AT-free graphs, a well-studied graph class that contains interval, permutation, and cocomparability graphs. Our result is the first non-trivial algorithm that shows fixed-parameter tractability of Bandwidth on a graph class on which the problem remains NP -complete.
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