The bandwidth minimization problem for caterpillars with hair length 3 is NP-complete
SIAM Journal on Algebraic and Discrete Methods
Computing the bandwidth of interval graphs
SIAM Journal on Discrete Mathematics
An optimal greedy heuristic to color interval graphs
Information Processing Letters
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
Triangulating graphs without asteroidal triples
Discrete Applied Mathematics
Pathwidth, Bandwidth, and Completion Problems to Proper Interval Graphs with Small Cliques
SIAM Journal on Computing
SIAM Journal on Discrete Mathematics
Graph classes: a survey
Computer Solution of Large Sparse Positive Definite
Computer Solution of Large Sparse Positive Definite
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Bandwidth of Split and Circular Permutation Graphs
WG '00 Proceedings of the 26th International Workshop on Graph-Theoretic Concepts in Computer Science
On Approximation Intractability of the Bandwidth Problem
On Approximation Intractability of the Bandwidth Problem
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Exact and Approximate Bandwidth
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Bandwidth of bipartite permutation graphs in polynomial time
Journal of Discrete Algorithms
Approximability of the path-distance-width for AT-free graphs
WG'11 Proceedings of the 37th international conference on Graph-Theoretic Concepts in Computer Science
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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} 驴V such that for every edge uv 驴 E, |β 驴 1(u) 驴 β 驴 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.