Bandwidth on AT-Free Graphs

Abstract

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.KeywordsInterval GraphGraph ClassHair LengthPermutation GraphVertex PairThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.