Graph bandwidth

In graph theory, the graph bandwidth problem is to label the n vertices v_i of a graph G with distinct integers ⁠${\displaystyle f(v_{i})}$⁠ so that the quantity ${\displaystyle \max\{\,|f(v_{i})-f(v_{j})|:v_{i}v_{j}\in E\,\}}$ 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 w_ij and the cost function to be minimized is ${\displaystyle \max\{\,w_{ij}|f(v_{i})-f(v_{j})|:v_{i}v_{j}\in E\,\}}$.

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 (Kaplan & Shamir 1996).

For fixed ${\displaystyle k}$ define for every ${\displaystyle i}$ the set ${\displaystyle I_{k}(i):=[i,i+k+1)}$. ${\displaystyle G_{k}(n)}$ is the corresponding interval graph formed from the intervals ${\displaystyle I_{k}(1),I_{k}(2),...I_{k}(n)}$. These are exactly the proper interval graphs of graphs having bandwidth ${\displaystyle k}$. These graphs called be cyclically interval graphs because the intervals can be assigned to layers ${\displaystyle L_{1},L_{2},...L_{k+1}}$ 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.

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

The bandwidth of a path graph P_n on n vertices is 1, and for a complete graph K_m we have ${\displaystyle \varphi (K_{n})=n-1}$. For the complete bipartite graph K_m,n,

${\displaystyle \varphi (K_{m,n})=\lfloor (m-1)/2\rfloor +n}$, assuming ${\displaystyle m\geq n\geq 1,}$

which was proved by Chvátal.^[3] As a special case of this formula, the star graph ${\displaystyle S_{k}=K_{k,1}}$ on k + 1 vertices has bandwidth ${\displaystyle \varphi (S_{k})=\lfloor (k-1)/2\rfloor +1}$.

For the hypercube graph ${\displaystyle Q_{n}}$ on ${\displaystyle 2^{n}}$ vertices the bandwidth was determined by Harper (1966) to be

${\displaystyle \varphi (Q_{n})=\sum _{m=0}^{n-1}{\binom {m}{\lfloor m/2\rfloor }}.}$

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

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,

${\displaystyle \varphi (G)\geq \chi (G)-1;}$

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

${\displaystyle \lceil (n-1)/\operatorname {diam} (G)\rceil \leq \varphi (G)\leq n-\operatorname {diam} (G),}$

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

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 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: Chung (1988) proved that if T is a tree of maximum degree at most ∆, then

${\displaystyle \varphi (T)\leq {\frac {5n}{\log _{\Delta }n}}.}$

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

${\displaystyle \varphi (G)\leq {\frac {20n}{\log _{\Delta }n}}.}$

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.^[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]

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.

• 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 (5): 1217–1227. arXiv:0910.3014. 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 (3): 223–254. doi:10.1002/jgt.3190060302.
• Chung, Fan R. K. (1988), "Labelings of Graphs", in Beineke, Lowell W.; Wilson, Robin J. (eds.), 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, arXiv:1012.1344, doi:10.4310/joc.2012.v3.n4.a5
• Harper, L. (1966). "Optimal numberings and isoperimetric problems on graphs". Journal of Combinatorial Theory. 1 (3): 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.