In graph theory, the graph bandwidth problem is to label the n vertices vi of a graph G with distinct integers f(vi) so that the quantity is minimized (E is the edge set of G). The problem may be visualized as placing the vertices of a graph at distinct integer points along the x-axis so that the length of the longest edge is minimized. Such placement is called linear graph arrangement, linear graph layout or linear graph placement.
The weighted graph bandwidth problem is a generalization wherein the edges are assigned weights wij and the cost function to be minimized is .
In terms of matrices, the (unweighted) graph bandwidth is the bandwidth of the symmetric matrix which is the adjacency matrix of the graph. The bandwidth may also be defined as one less than the maximum clique size in a proper interval supergraph of the given graph, chosen to minimize its clique size (Kaplan & Shamir 1996).
Bandwidth formulas for some graphs
For several families of graphs, the bandwidth is given by an explicit formula.
- , assuming
The bandwidth of a graph can be bounded in terms of various other graph parameters. For instance, letting χ(G) denote the chromatic number of G,
where is the number of vertices in .
If a graph G has bandwidth k, then its pathwidth is at most k (Kaplan & Shamir 1996), and its tree-depth is at most k log(n/k) (Gruber 2012). In contrast, as noted in the previous section, the star graph Sk, a structurally very simple example of a tree, has comparatively large bandwidth. Observe that the pathwidth of Sk is 1, and its tree-depth is 2.
Some graph families of bounded degree have sublinear bandwidth: Chung (1988) proved that if T is a tree of maximum degree at most ∆, then
Computing the bandwidth
Both the unweighted and weighted versions are special cases of the quadratic bottleneck assignment problem. The bandwidth problem is NP-hard, even for some special cases. Regarding the existence of efficient approximation algorithms, it is known that the bandwidth is NP-hard to approximate within any constant, and this even holds when the input graphs are restricted to caterpillar trees (Dubey, Feige & Unger 2010). For the case of dense graphs, a 3-approximation algorithm was designed by Karpinski, Wirtgen & Zelikovsky (1997). On the other hand, a number of polynomially-solvable special cases are known. A heuristic algorithm for obtaining linear graph layouts of low bandwidth is the Cuthill–McKee algorithm. Fast multilevel algorithm for graph bandwidth computation was proposed in.
The interest in this problem comes from some application areas.
Another application domain is in electronic design automation. In standard cell design methodology, typically standard cells have the same height, and their placement is arranged in a number of rows. In this context, graph bandwidth problem models the problem of placement of a set of standard cells in a singe row with the goal of minimizing the maximal propagation delay (which is assumed to be proportional to wire length).
- Pathwidth, a different NP-complete optimization problem involving linear layouts of graphs.
- (Chinn et al. 1982)
- "Coping with the NP-Hardness of the Graph Bandwidth Problem", Uriel Feige, Lecture Notes in Computer Science, Volume 1851, 2000, pp. 129-145, doi:10.1007/3-540-44985-X_2
- A remark on a problem of Harary. V. Chvátal, Czechoslovak Mathematical Journal 20(1):109–111, 1970. http://dml.cz/dmlcz/100949
- Optimal Labelling of a product of two paths. J. Chvatálová, Discrete Mathematics 11, 249–253, 1975.
- Chinn et al. 1982
- Garey–Johnson: GT40
- Ilya Safro and Dorit Ron and Achi Brandt (2008). "Multilevel Algorithms for Linear Ordering Problems". ACM Journal of Experimental Algorithmics 13: 1.4–1.20. doi:10.1145/1412228.1412232.
- Böttcher, J.; Pruessmann, K. P.; Taraz, A.; Würfl, A. (2010). "Bandwidth, expansion, treewidth, separators and universality for bounded-degree graphs". European Journal of Combinatorics 31: 1217–1227. doi:10.1016/j.ejc.2009.10.010.
- Chinn, P. Z.; Chvátalová, J.; Dewdney, A. K.; Gibbs, N. E. (1982). "The bandwidth problem for graphs and matrices—a survey". Journal of Graph Theory 6: 223–254. doi:10.1002/jgt.3190060302.
- Chung, Fan R. K. (1988), "Labelings of Graphs", in Beineke, Lowell W.; Wilson, Robin J., Selected Topics in Graph Theory (PDF), Academic Press, pp. 151–168, ISBN 978-0-12-086203-0
- Dubey, C.; Feige, U.; Unger, W. (2010). "Hardness results for approximating the bandwidth". Journal of Computer and System Sciences 77: 62–90. doi:10.1016/j.jcss.2010.06.006.
- Garey, M.R.; Johnson, D.S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. New York: W.H. Freeman. ISBN 0-7167-1045-5.
- Gruber, Hermann (2012), "On Balanced Separators, Treewidth, and Cycle Rank", Journal of Combinatorics 3 (4): 669–682, doi:10.4310/joc.2012.v3.n4.a5
- Harper, L. (1966). "Optimal numberings and isoperimetric problems on graphs". Journal of Combinatorial Theory 1: 385–393. doi:10.1016/S0021-9800(66)80059-5.
- Kaplan, Haim; Shamir, Ron (1996), "Pathwidth, bandwidth, and completion problems to proper interval graphs with small cliques", SIAM Journal on Computing 25 (3): 540–561, doi:10.1137/s0097539793258143
- Karpinski, Marek; Wirtgen, Jürgen; Zelikovsky, Aleksandr (1997). "An Approximation Algorithm for the Bandwidth Problem on Dense Graphs". Electronic Colloquium on Computational Complexity 4 (17).
- Minimum bandwidth problem, in: Pierluigi Crescenzi and Viggo Kann (eds.), A compendium of NP optimization problems. Accessed May 26, 2010.