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Graph bandwidth

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In graph theory, the graph bandwidth problem is to label the n vertices vi of a graph G with distinct integers so that the quantity is minimized (E is the edge set of G).[1] 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.[2]

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 minimal bandwidth of a symmetric matrix which is an 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).

Cyclically interval graphs

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For fixed define for every the set . is the corresponding interval graph formed from the intervals . These are exactly the proper interval graphs of graphs having bandwidth . These graphs called be cyclically interval graphs because the intervals can be assigned to layers in cyclically order, where the intervals of a layer doesn't intersect.

Therefore, one see the relation to the pathwidth. Pathwidth restricted graphs are minor closed but the set of subgraphs of cyclically interval graphs are not. This follows from the fact, that thrinking degree 2 vertices may increase the bandwidth.

Alternate adding vertices on edges can decrease the bandwidth. Hint: The last problem is known as topologic bandwidth. An other graph measure related through the bandwidth is the bisection bandwidth.

Bandwidth formulas for some graphs

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For several families of graphs, the bandwidth is given by an explicit formula.

The bandwidth of a path graph Pn on n vertices is 1, and for a complete graph Km we have . For the complete bipartite graph Km,n,

, assuming

which was proved by Chvátal.[3] As a special case of this formula, the star graph on k + 1 vertices has bandwidth .

For the hypercube graph on vertices the bandwidth was determined by Harper (1966) to be

Chvatálová showed[4] that the bandwidth of the m × n square grid graph , that is, the Cartesian product of two path graphs on and vertices, is equal to min{m,n}.

Bounds

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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,

letting diam(G) denote the diameter of G, the following inequalities hold:[5]

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

More generally, for planar graphs of bounded maximum degree at most , a similar bound holds (cf. Böttcher et al. 2010):

Computing the bandwidth

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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.[6] 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 with maximum hair length 2 (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.[2] 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.[7]

Applications

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The interest in this problem comes from some application areas.

One area is sparse matrix/band matrix handling, and general algorithms from this area, such as Cuthill–McKee algorithm, may be applied to find approximate solutions for the graph bandwidth problem.

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 single row with the goal of minimizing the maximal propagation delay (which is assumed to be proportional to wire length).

See also

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  • Cutwidth and pathwidth, different NP-complete optimization problems involving linear layouts of graphs.

References

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  1. ^ (Chinn et al. 1982)
  2. ^ a b "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
  3. ^ A remark on a problem of Harary. V. Chvátal, Czechoslovak Mathematical Journal 20(1):109–111, 1970. http://dml.cz/dmlcz/100949
  4. ^ Optimal Labelling of a product of two paths. J. Chvatálová, Discrete Mathematics 11, 249–253, 1975.
  5. ^ Chinn et al. 1982
  6. ^ Garey–Johnson: GT40
  7. ^ 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.
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  • Minimum bandwidth problem, in: Pierluigi Crescenzi and Viggo Kann (eds.), A compendium of NP optimization problems. Accessed May 26, 2010.