Years and Authors of Summarized Original Work
1998; Feige
2000; Feige
Problem Definition
The graph bandwidth problem concerns producing a linear ordering of the vertices of a graph \( { G = (V,E) } \) so as to minimize the maximum “stretch” of any edge in the ordering. Formally, let \( { n = |V| } \), and consider any one-to-one mapping \( { \pi : V \to \{1,2, \dots, n\} } \). The bandwidth of this ordering is \( \mathsf{bw}_{\pi}(G) = \max_{\{u,v\} \in E} |\pi(u) \)\( -\pi(v)| \). The bandwidth of G is given by the bandwidth of the best possible ordering: \( { \mathsf{bw}(G) = \min_{\pi} \mathsf{bw}_{\pi}(G) } \).
The original motivation for this problem lies in the preprocessing of sparse symmetric square matrices. Let A be such an \( { n \times n } \) matrix, and consider the problem of finding a permutation matrix P such that the non-zero entries of \( { P^{\text{T}} A P } \)all lie in as narrow a band as possible about the diagonal. This problem is equivalent to minimizing the...
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Blum A, Konjevod G, Ravi R, Vempala S (2000) Semi-definite relaxations for minimum bandwidth and other vertex-ordering problems. Theor Comput Sci 235(1):25–42, Selected papers in honor of Manuel Blum (Hong Kong, 1998)
Bourgain J (1985) On Lipschitz embedding of finitemetric spaces in Hilbert space. Israel J Math 52(1–2):46–52
Chinn PZ, Chvátalová J, Dewdney AK, Gibbs NE (1982) The bandwidth problem for graphs and matrices – a survey. J Graph Theory 6(3):223–254
Chung FRK, Seymour PD (1989) Graphs withsmall bandwidth and cutwidth. Discret Math 75(1–3):113–119, Graph theory and combinatorics, Cambridge (1988)
Dunagan J, Vempala S (2001) On Euclidean embeddings and bandwidth minimization. In: Randomization, approximation, and combinatorial optimization. Springer, pp 229–240
Even G, Naor J, Rao S, Schieber B (2000) Divide-and-conquer approximation algorithms via spreading metrics. J ACM 47(4):585–616
Feige U (2000) Approximating the bandwidth via volume respecting embeddings. J Comput Syst Sci 60(3):510–539
George A, Liu JWH (1981) Computer solution of large sparse positive definite systems. Prentice-hall series in computational mathematics. Prentice-Hall, Englewood Cliffs
Gupta A (2001) Improved bandwidth approximation for trees and chordal graphs. J Algorithms 40(1):24–36
Krauthgamer R, Lee JR, Mendel M, Naor A (2005) Measured descent: a new embedding method for finite metrics. Geom Funct Anal 15(4):839–858
Krauthgamer R, Linial N, Magen A (2004) Metric embeddings – beyond one-dimensional distortion. Discret Comput Geom 31(3):339–356
Lee JR (2006) Volume distortion for subsets of Euclidean spaces. In: Proceedings of the 22nd annual symposium on computational geometry. ACM, Sedona, pp 207–216
Linial N (2002) Finite metric-spaces – combinatorics, geometry and algorithms. In: Proceedings of the international congress of mathematicians, vol. III, Beijing, 2002. Higher Ed. Press, Beijing, pp 573–586
Linial N, London E, Rabinovich Y (1995) The geometry of graphs and some of its algorithmic applications. Combinatorica 15(2):215–245
Papadimitriou CH (1976) The NP-completeness of the bandwidth minimization problem. Computing 16(3):263–270
Rao S (1999) Small distortion and volume preserving embeddings for planar and Euclidean metrics. In: Proceedings of the 15th annual symposium on computational geometry. ACM, New York, pp 300–306
Strang G (1980) Linear algebra and its applications, 2nd edn. Academic [Harcourt Brace Jovanovich Publishers], New York
Unger W (1998) The complexity of the approximation of the bandwidth problem. In: 39th annual symposium on foundations of computer science. IEEE, 8–11 Oct 1998, pp 82–91
Vempala S (1998) Random projection: a new approach to VLSI layout. In: 39th annual symposium on foundations of computer science. IEEE, 8–11 Oct 1998, pp 389–398
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Lee, J. (2016). Graph Bandwidth. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_169
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