Bandwidth on AT-Free Graphs

We study the classical Bandwidth problem from the viewpoint of parameterized algorithms. In the Bandwidth problem we are given a graph G = (V,E) together with a positive integer k, and asked whether there is an bijective function β: {1, ..., n} ?V such that for every edge uv ? E, |β ? 1(u) ? β ? 1(v)| ≤ k. The problem is notoriously hard, and it is known to be NP-complete even on very restricted subclasses of trees. The best known algorithm for Bandwidth for small values of k is the celebrated algorithm by Saxe [SIAM Journal on Algebraic and Discrete Methods, 1980 ], which runs in time $2^{{\mathcal{O}}(k)}n^{k+1}$. In a seminal paper, Bodlaender, Fellows and Hallet [STOC 1994 ] ruled out the existence of an algorithm with running time of the form $f(k)n^{{\mathcal{O}}(1)}$ for any function f even for trees, unless the entire W-hierarchy collapses. We initiate the search for classes of graphs where Bandwidth is fixed parameter tractable (FPT), that is, solvable in time $f(k)n^{{\mathcal{O}}(1)}$ for some function f. In this paper we present an algorithm with running time $2^{{\mathcal O}(k \log k)} n^2$ 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 FPT algorithm for Bandwidth on a graph class where the problem remains NP-complete.