Graph bandwidth

In graph theory, the graph bandwidth problem may be visualized as placing the vertices of a given graph at distinct integer positions along the number line so that the length of the longest edge is minimized. Such placement is called linear graph arrangement, linear graph layout or linear graph placement.^[1] It may be formalized as labeling the n {\displaystyle n} vertices v i {\displaystyle v_{i}} of a graph G {\displaystyle G} with distinct integers f ( v i ) {\displaystyle f(v_{i})} so that the quantity max { | f ( v i ) − f ( v j ) | : v i v j ∈ E } {\displaystyle \max\{\,|f(v_{i})-f(v_{j})|:v_{i}v_{j}\in E\,\}} is minimized, where E {\displaystyle E} is the edge set of G {\displaystyle G} .^[2]

The weighted graph bandwidth problem is a generalization wherein the edges are assigned weights w i j {\displaystyle w_{ij}} and the cost function to be minimized is the product of weight with length, max { w i j | f ( v i ) − f ( v j ) | : v i v j ∈ E } {\displaystyle \max\{\,w_{ij}|f(v_{i})-f(v_{j})|:v_{i}v_{j}\in E\,\}} .

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.^[3]

For several families of graphs, the bandwidth φ ( G ) {\displaystyle \varphi (G)} is given by an explicit formula.

The bandwidth of a path graph P n {\displaystyle P_{n}} on n {\displaystyle n} vertices is 1, and the bandwidth of a complete graph K m {\displaystyle K_{m}} is φ ( K n ) = n − 1 {\displaystyle \varphi (K_{n})=n-1} . For the complete bipartite graph K m , n {\displaystyle K_{m,n}} , φ ( K m , n ) = ⌊ ( m − 1 ) / 2 ⌋ + n , {\displaystyle \varphi (K_{m,n})=\lfloor (m-1)/2\rfloor +n,} assuming m ≥ n ≥ 1 {\displaystyle m\geq n\geq 1} . As a special case of this formula, the star graph S k = K k , 1 {\displaystyle S_{k}=K_{k,1}} on k + 1 {\displaystyle k+1} vertices has bandwidth φ ( S k ) = ⌊ ( k − 1 ) / 2 ⌋ + 1 {\displaystyle \varphi (S_{k})=\lfloor (k-1)/2\rfloor +1} .^[4]

For the hypercube graph Q n {\displaystyle Q_{n}} on 2 n {\displaystyle 2^{n}} vertices the bandwidth is^[5] φ ( Q n ) = ∑ m = 0 n − 1 ( m ⌊ m / 2 ⌋ ) . {\displaystyle \varphi (Q_{n})=\sum _{m=0}^{n-1}{\binom {m}{\lfloor m/2\rfloor }}.}

The bandwidth of the m × n {\displaystyle m\times n} square grid graph P m × P n {\displaystyle P_{m}\times P_{n}} , that is, the Cartesian product of two path graphs on m {\displaystyle m} and n {\displaystyle n} vertices, is equal to min { m , n } {\displaystyle \min\{m,n\}} .^[6]

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,

φ ( G ) ≥ χ ( G ) − 1 ; {\displaystyle \varphi (G)\geq \chi (G)-1;}

letting diam(G) denote the diameter of G, the following inequalities hold:^[2]

⌈ ( n − 1 ) / diam ⁡ ( G ) ⌉ ≤ φ ( G ) ≤ n − diam ⁡ ( G ) , {\displaystyle \lceil (n-1)/\operatorname {diam} (G)\rceil \leq \varphi (G)\leq n-\operatorname {diam} (G),}

where n {\displaystyle n} is the number of vertices in G {\displaystyle G} .

If a graph G has bandwidth k, then its pathwidth is at most k,^[3] and its tree-depth is at most k log(n/k).^[7] In contrast, as noted in the previous section, the star graph S_k, a structurally very simple example of a tree, has comparatively large bandwidth. Observe that the pathwidth of S_k is 1, and its tree-depth is 2.

Some graph families of bounded degree have sublinear bandwidth: if T is a tree of maximum degree at most ∆, then ^[8]

φ ( T ) ≤ 5 n log Δ ⁡ n . {\displaystyle \varphi (T)\leq {\frac {5n}{\log _{\Delta }n}}.}

More generally, for planar graphs of bounded maximum degree at most ∆, a similar bound holds:^[9]

φ ( G ) ≤ 20 n log Δ ⁡ n . {\displaystyle \varphi (G)\leq {\frac {20n}{\log _{\Delta }n}}.}

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.^[10] 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.^[11] For arbitrary graphs with n {\displaystyle n} vertices, the best approximation ratio known is O ( log 3 ⁡ n log ⁡ log ⁡ n ) {\displaystyle O(\log ^{3}n{\sqrt {\log \log n}})} , using semidefinite programming.^[12] For the case of dense graphs, a 3-approximation algorithm is known.^[13] On the other hand, a number of polynomially-solvable special cases are known.^[1] 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.^[14]

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).

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

• Minimum bandwidth problem, in: Pierluigi Crescenzi and Viggo Kann (eds.), A compendium of NP optimization problems. Accessed May 26, 2010.