The bandwidth minimization problem for caterpillars with hair length 3 is NP-complete
SIAM Journal on Algebraic and Discrete Methods
Graphs with small bandwidth and cutwidth
Discrete Mathematics
Improved bandwidth approximation for trees
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Approximating the bandwidth via volume respecting embeddings
Journal of Computer and System Sciences - 30th annual ACM symposium on theory of computing
Coping with the NP-Hardness of the Graph Bandwidth Problem
SWAT '00 Proceedings of the 7th Scandinavian Workshop on Algorithm Theory
On Euclidean Embeddings and Bandwidth Minimization
APPROX '01/RANDOM '01 Proceedings of the 4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and 5th International Workshop on Randomization and Approximation Techniques in Computer Science: Approximation, Randomization and Combinatorial Optimization
The Complexity of the Approximation of the Bandwidth Problem
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
On Approximation Intractability of the Bandwidth Problem
On Approximation Intractability of the Bandwidth Problem
Measured Descent: A New Embedding Method for Finite Metrics
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Confronting hardness using a hybrid approach
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Labeling schemes for weighted dynamic trees
Information and Computation
Dynamic routing schemes for graphs with low local density
ACM Transactions on Algorithms (TALG)
Bandwidth of bipartite permutation graphs in polynomial time
Journal of Discrete Algorithms
Bandwidth of bipartite permutation graphs in polynomial time
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
Hardness results for approximating the bandwidth
Journal of Computer and System Sciences
Low Distortion Maps Between Point Sets
SIAM Journal on Computing
Capacitated domination faster than O(2n)
SWAT'10 Proceedings of the 12th Scandinavian conference on Algorithm Theory
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A caterpillar is a tree in which all vertices of degree three or more lie on one path, called the backbone. We present a polynomial time algorithm that produces a linear arrangement of the vertices of a caterpillar with bandwidth at most O(log n/loglog n) times the local density of the caterpillar, where the local density is a well known lower bound on the bandwidth. This result is best possible in the sense that there are caterpillars whose bandwidth is larger than their local density by a factor of Ω(log n/loglog n). The previous best approximation ratio for the bandwidth of caterpillars was O(log n). We show that any further improvement in the approximation ratio would require using linear arrangements that do not respect the order of the vertices of the backbone. We also show how to obtain a (1 + ε) approximation for the bandwidth of caterpillars in time $2^{\tilde{O}(\sqrt{n/\epsilon})}$. This result generalizes to trees, planar graphs, and any family of graphs with treewidth $\tilde{O}(\sqrt{n})$.