M. R. Garey, R. L. Graham, D. S. Johnson, and D. E. Knuth, Complexity results for bandwidth minimization, SIAM J. Appl. Math., 34 (1978), pp. 477--495.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....For any graph G, ts(G) d (G) 2e. For trees of maximum degree at most 3 it is easy to prove that ts(G) d (G) 2e. It is an interesting question whether treespan can be computed in polynomial time for trees of larger max degree. Notice that bandwidth remains NP complete on trees of max degree 3 [13]. ....

M. R. Garey, R. L. Graham, D. S. Johnson, and D. E. Knuth, Complexity results for bandwidth minimization, SIAM J. Appl. Math., 34 (1978), pp. 477-495.

.... Gamma (v)j. The bandwidth of G is defined by OE(G) min OE (G) The bandwidth minimization problem (for graphs) is to determine for a graph G and an integer k if OE(G) k. Papadimitriou [9] first showed that the bandwidth minimization problem is NP complete. Garey, Graham, Johnson and Knuth [7] later strengthened this result, showing that the problem remains NP complete when restricted to free binary trees. Several heuristic algorithms for bandwidth minimization were proposed in the late sixties and early seventies. More recently, Saxe [10] has found a dynamic programming algorithm ....

....x precedes y in V i . The running time of MLA1 is dominated by the count gc and count gp functions. A straightforward implementation of these gives a running time of O(n ) The procedure make mod levels can be implemented to run in O(jEj) O(n ) time, and the sorting steps in lines [6] and [7] require at most O(n log n) There are other possible strategies for arranging the vertices within each level. Cuthill and McKee [5] who first suggested the level algorithms, arranged the vertices within levels according to the order in which they were visited by a breadth first search ....

Garey, Michael R., R. L. Graham, David S. Johnson, D. E. Knuth. "Complexity Results for Bandwidth Minimization". In SIAM Journal of Applied Mathematics 34 , 477-495, 5/78.

....layouts of G, namely, bw(G) minfbwL (G) j L is a layout of Gg. The BANDWIDTH problem is to decide for a given graph G and integer k, if bw(G) k. This problem has been studied intensely because of its application to sparse matrix algebra [9] It is known to be NP complete even for binary trees [16] and for caterpillars with hair length at most three [34] On the other hand, it is solvable in O(n ) for arbitrary k [23] and in linear time for k = 2 [16] Sandwich Problems: Given two graphs G ) such that G a supergraph of G , a graph G = V; E) is called a sandwich for ....

.... has been studied intensely because of its application to sparse matrix algebra [9] It is known to be NP complete even for binary trees [16] and for caterpillars with hair length at most three [34] On the other hand, it is solvable in O(n ) for arbitrary k [23] and in linear time for k = 2 [16]. Sandwich Problems: Given two graphs G ) such that G a supergraph of G , a graph G = V; E) is called a sandwich for this pair if E E E . G must be sandwiched between G , hence the name) Let E be the set of all edges in the complete graph with vertex set V ....

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M. R. Garey, R. L. Graham, D. S. Johnson, and D. E. Knuth. Complexity results for bandwidth minimization. SIAM J. Appl. Math., 34(3):477--495, 1978.

....If G = V; E) is a cocomparability graph of maximum degree (G) 3, then bw(G) b c. We will now turn our attention to Problem (iii) It is well known that graphs with bandwidth at most 2 can be recognized in linear time and that a 2 labeling of such graphs can also be found in linear time [6], 9] 1] In view of Theorem 1, this implies that posets with linear discrepancy at most 2 can be recognized in linear time. Furthermore, we will describe now how the proof of Theorem 1 in [5] implies that a linear extension of uncertainty at most 2 of such posets can be found in polynomial ....

M.R. Garey, R.L. Graham, D.S. Johnson and D.E. Knuth, Complexity results for bandwidth minimization, SIAM J. Appl. Math. 34 (1978), 477-495.

....constraint graph can be used as a branching order that reduces backtracking. For a survey on the bandwidth problem and early approaches, see [2] The minimum bandwidth problem was first shown to be NP hard in 1976 [15] and later even for trees of degree at most three and for caterpillars [7, 13]. Approximations algorithms have been known only for some special families of graphs, such as caterpillars or asteroidal triple free graphs [11, 12] 2 The Semi Definite Relaxation Our approximation algorithm begins with an SDP (Semi Definite Programming) relaxation. First we motivate and ....

M Garey, R. Graham, D. Johnson, D. E. Knuth, Complexity results for bandwidth minimization, SIAM J. Appl. Math. 34:477 495, 1978.

....of the paper advertised in the abstract, and stated more formally below is interesting on two counts. First, it provides a rare example of a natural question about trees which is unlikely to be polynomial time solvable. Two other examples are determining a vertex ordering of minimum bandwidth [1, 4], or determining the harmonious chromatic number [2] Second, it is, as far as we are aware, the first intractability result concerning the counting of unlabelled structures. Some definitions. By rooted tree (T;r) we simply mean a tree T with a distinguished vertex r, the root. An embedding of ....

M. R. Garey, R. L. Graham, D. S. Johnson and D. E. Knuth, Complexity results for bandwidth minimization, SIAM Journal on Applied Mathematics 34 (1978), 477--495.

....above such that for every vertex v its numbering is greater than any numbering of a vertex u such that (u; v) 2 E. The DBANDWIDTH problem corresponds to that of minimizing the bandwidth of an upper triangular matrix by simultaneous row and column permutations (cf. Garey, Graham, Johnson and Knuth [GGJK78]) The problem is known to be NP hard even if restricted to binary trees (cf. GGJK78] or caterpillars with hairs of length at most 3, Monien [M83] This makes the BANDWIDTH one of the very rare combinatorial problems which are computationally hard for trees. Interestingly, the problem is ....

....u such that (u; v) 2 E. The DBANDWIDTH problem corresponds to that of minimizing the bandwidth of an upper triangular matrix by simultaneous row and column permutations (cf. Garey, Graham, Johnson and Knuth [GGJK78] The problem is known to be NP hard even if restricted to binary trees (cf. [GGJK78]) or caterpillars with hairs of length at most 3, Monien [M83] This makes the BANDWIDTH one of the very rare combinatorial problems which are computationally hard for trees. Interestingly, the problem is efficiently computable for complete trees, Smithline [Sm95] Only a very few special cases ....

M. Garey, R. Graham, D. Johnson, D. Knuth, Complexity Results for Bandwidth Minimization, SIAM J. Appl. Math. 34 (1978), pp. 477-495.

....both heuristic and exact algorithms is also reported. As the objective of the bandwidth minimization is to preprocess data in order to speed up the solution of large systems, fast algorithms are required. In this respect, it was shown more than 20 years ago by Garey, Graham, Johnson and Knuth [3] that graphs (or matrices) of bandwidth two can be recognized in linear time. However, the algorithm presented in [3] exploits very few properties of such graphs, and, as a result, it requires a very complex case analysis which makes it dicult to follow in detail and to implement. For the special ....

....data in order to speed up the solution of large systems, fast algorithms are required. In this respect, it was shown more than 20 years ago by Garey, Graham, Johnson and Knuth [3] that graphs (or matrices) of bandwidth two can be recognized in linear time. However, the algorithm presented in [3] exploits very few properties of such graphs, and, as a result, it requires a very complex case analysis which makes it dicult to follow in detail and to implement. For the special case of bandwidth 2 biconnected graphs, a complete characterization was presented more recently by Makedon, Sheinwald ....

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M.R. Garey, R.L. Graham, D.S. Johnson and D.E. Knuth, \Complexity Results for Bandwidth Minimization", SIAM Journal on Applied Mathematics 34 (1978) 477-495.

.... with interval graphs, of importance to both algorithmic and structural graph theory, and with applications to VLSI layout [6, 11] Bandwidth is another widely studied graph parameter, with applications to sparse matrices [1] and notorious for the difficulty of its computation even for trees [3]. Recently, in a study of problems motivated by molecular biology, Kaplan and Shamir [5] showed a somewhat surprising connection between bandwidth and pathwidth, using the well known notion of proper interval graphs, or equivalently unit interval graphs. A similar connection was shown by [2] based ....

M. Garey, R. Graham, D. Johnson and D. Knuth, Complexity results for bandwidth minimization, SIAM J. Applied Math, vol. 34. no. 3, May 1978.

....complexity remains open. Positive results. Recall that NP completeness results do not rule out the existence of e#cient algorithms to get optimal layouts on particular classes of graphs. Table 4 summarizes 6 Problem NP complete Ref. Bandwidth in general [83] for trees with maximum degree 3 [37] for caterpillars with hairlength # 3 [76] for caterpillars with # 1 hair per backbone node [76] for grid graphs and unit disk graphs [26] MinLA in general [39] for bipartite graphs [38] Cutwidth in general [40] for graphs with maximum degree 3 [73] for planar graphs with maximum degree 3 ....

....[77] for chordal graphs [45] for grid graphs and unit disk graphs [26] for bipartite graphs [42] SumCut in general [24, 64] for cobipartite graphs [101] EdgeBis in general [39] Table 2: Review of NP completeness results for graph layout problems. Problem Complexity Ref. Bandwidth(2) O(n) [37] Cutwidth(2) O(n) 37] Cutwidth(k) O(n 2 ) 35] ModCut(k) O(n 2 ) 35] VertSep(k) O(n) 14] Table 3: Complexity results for layout problems with fixed parameter (n denotes the number of nodes in the graph and k the parameter) 7 Problem Class of graph Complexity Ref. Bandwidth ....

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M. R. Garey, R. L. Graham, D. S. Johnson, and D. Knuth. Complexity results for bandwidth minimization. SIAM Journal on Applied Mathematics, 34:477--495, 1978.

....= #(u) #(v) The complexity of a graph in terms of a linear layout is usually obtained by measuring length, crossing edges or neighbors placement. The bandwidth problem asks for a layout minimizing the maximum edge length. The problem is NP complete [28] even for trees with maximum degree 3 [14] or caterpillars with hair length 3 [25] It can be approximated within a constant for some restricted classes of trees [18] but has no polynomial time approximation scheme for trees [4] It has a constant randomized approximation algorithm for dense instances [21] and no polynomial time ....

M. R. Garey, R. L. Graham, D. S. Johnson, and D. Knuth. Complexity results for bandwidth minimization. SIAM J on Applied Mathematics, 34:477--495, September 1978.

....34, 59, 54] NP C for bipartite graphs [30] NC for trees [19] O(logn) approximable [69] O(log log n) approx. for planar graphs [69] PTAS for dense graphs [5] SumCut NP C [30] P for trees [23] NC for trees [23] Bandwidth NP C [62] APX for certain trees [33] NP C for trees # = 3 [29], APX for dense graphs [46] caterpillars with hair length 3 [55] no PTAS for trees [9] no APX in general [45] Cutwidth NP C [32] P for trees [75] NP C for pl. graphs # = 3 [57] NC for trees [19] APX for dense graphs [5] VertSep NP C [52] P for trees [28] EdgeBis NP C [31] P ....

M. R. Garey, R. L. Graham, D. S. Johnson, and D. Knuth. Complexity results for bandwidth minimization. SIAM Journal on Applied Mathematics, 34:477--495, September 1978.

....there is a permutation matrix P such that P Delta A Delta P T is a matrix with all nonzero entries on the main diagonal or on the k diagonals on either side of this main diagonal. Computing the bandwidth of a graph is NP complete [14] even when restricted to trees of maximum degree three [7]. There are only a few graph classes known for which the bandwidth can be computed efficiently. Such graph classes are the class of theta graphs [15] cographs [10] and chain graphs. Another one is the class of caterpillars with hairs of length one and two [1] However, for caterpillars with ....

....[10] and chain graphs. Another one is the class of caterpillars with hairs of length one and two [1] However, for caterpillars with hairs of length at most three, the bandwidth problem remains NP complete [13] It can be checked in linear time whether the bandwidth of a graph is at most two [7]. For general k, there is an O(n k ) algorithm to check whether the bandwidth of a graph is at most k [8] In some sense, this is best possible, since it was shown in [4] that bandwidth is W [t] hard for all t in the fixed parameter hierarchy. Hence in general it is not expected that there is an ....

M.R. Garey, R.L. Graham, D.S. Johnson and D.E. Knuth, Complexity results for bandwidth minimization, SIAM Journal on Applied Mathematics 34 (1978), pp. 477--495.

....are given in [8, 3] Supported by NSF grants CCR 9505448 and CCR9820951. In 1976, this problem was shown to be NP hard for general graphs by Papadimitriou [14] Subsequent work strengthened the hardness result to trees with maximum degree 3, and to caterpillars of hair length at most 3 [5, 13], making this one of the few problems known to be hard even when the input graphs are trees of a very simple form. Furthermore, it has also been shown that the Bandwidth Minimization problem is hard to even approximate on trees. In fact, it is NP hard to approximate it to within any constant even ....

Michael R. Garey, Ronald L. Graham, David S. Johnson, and Donald E. Knuth. Complexity results for bandwidth minimization. SIAM J. Appl. Math., 34(3):477-495, 1978.

....of the paper advertised in the abstract, and stated more formally below is interesting on two counts. First, it provides a rare example of a natural question about trees which is unlikely to be polynomial time solvable. Two other examples are determining a vertex ordering of minimum bandwidth [1, 4], or determining the harmonious chromatic number [2] Second, it is, as far as we are aware, the first intractability result concerning the counting of unlabelled structures. Some definitions. By rooted tree (T;r) we simply mean a tree T with a distinguished vertex r, the root. An embedding of ....

M. R. Garey, R. L. Graham, D. S. Johnson and D. E. Knuth, Complexity results for bandwidth minimization, SIAM Journal on Applied Mathematics 34 (1978), 477--495.

....two can be done in linear time. D. The parametric version of Problem A is W [t] hard for any t 0. E. Problem B is NP complete and its parametric version is W[t] hard for every t 0. These results follow immediately from known results on bandwidth, using Theorem 3. 2: Result A follows from [11, 20], B follows from [14] C from [11] and D from [4] Result E follows since Problem B is a generalization of Problem A and thus at least as hard. We next describe a polynomial algorithm for Problem B when k is fixed. A sandwich instance is a triplet S = V; E r ; E f ) where V is a set of ....

....The parametric version of Problem A is W [t] hard for any t 0. E. Problem B is NP complete and its parametric version is W[t] hard for every t 0. These results follow immediately from known results on bandwidth, using Theorem 3. 2: Result A follows from [11, 20] B follows from [14] C from [11] and D from [4] Result E follows since Problem B is a generalization of Problem A and thus at least as hard. We next describe a polynomial algorithm for Problem B when k is fixed. A sandwich instance is a triplet S = V; E r ; E f ) where V is a set of vertices and E r ; E f are ....

M. R. Garey, R. L. Graham, D. S. Johnson, and D. E. Knuth. Complexity results for bandwidth minimization. SIAM J. Appl. Math., 34(3):477-- 495, 1978.

....of the paper advertised in the abstract, and stated more formally below is interesting on two counts. First, it provides a rare example of a natural question about trees which is unlikely to be polynomial time solvable. Two other examples are determining a vertex ordering of minimum bandwidth [1, 4], or determining the harmonious chromatic number [2] Second, it is, as far as we are aware, the rst intractability result concerning the counting of unlabelled structures. Some de nitions. By rooted tree (T; r) we simply mean a tree T with a distinguished vertex r, the root. An embedding of ....

M. R. Garey, R. L. Graham, D. S. Johnson and D. E. Knuth, Complexity results for bandwidth minimization, SIAM Journal on Applied Mathematics 34 (1978), 477-495.

....then the reordering problem is the minimum bandwidth problem for the resulting graph. For a survey on the bandwidth problem and early approaches, see [1] The minimum bandwidth problem was first shown to be NPhard in 1976 [10] and later even for trees of degree at most three and for caterpillars [5, 9]. Approximations algorithms have been known only for some special families of graphs, such as caterpillars or asteroidal triple free graphs [7, 8] 2 The Semi Definite Relaxation Our approximation algorithm begins with an SDP (Semi Definite Programming) relaxation. First we motivate and describe ....

M Garey, R. Graham, D. Johnson, D. E. Knuth, Complexity results for bandwidth minimization, SIAM J. Appl. Math. 34: 477--495, 1978.

....[6, 12, 29] for more information. Incumbent of the Joseph and Celia Reskin Career Development Chair. 1 On special families of graphs, the bandwidth can be computed in polynomial time [1, 17] However, computing the bandwidth on general graphs is NP hard [25] even on some subfamilies of trees [11, 24]. Deciding whether the bandwidth of a graph is at most 2 can be done in linear time [11] For any fixed value k, deciding whether the bandwidth is at most k can be done in time O(n k ) 28, 14] but apparently not much faster, as the problem is hard for any fixed level of the fixed parameter ....

....Chair. 1 On special families of graphs, the bandwidth can be computed in polynomial time [1, 17] However, computing the bandwidth on general graphs is NP hard [25] even on some subfamilies of trees [11, 24] Deciding whether the bandwidth of a graph is at most 2 can be done in linear time [11]. For any fixed value k, deciding whether the bandwidth is at most k can be done in time O(n k ) 28, 14] but apparently not much faster, as the problem is hard for any fixed level of the fixed parameter tractability hierarchy, even for trees [4] In practice, heuristics for minimizing the ....

M. Garey, R. Graham, D. Johnson, D. Knuth. "Complexity results for bandwidth minimization". SIAM J. Appl. Math. 34 (1978), 477-495.

....layouts of G, namely, bw(G) minfbwL (G) j L is a layout of Gg. The BANDWIDTH problem is to decide for a given graph G and integer k, if bw(G) k. This problem has been studied intensely because of its application to sparse matrix algebra [9] It is known to be NP complete even for binary trees [16] and for caterpillars with hair length at most three [34] On the other hand, it is solvable in O(n k ) for arbitrary k [23] and in linear time for k = 2 [16] Sandwich Problems: Given two graphs G 1 = V; E 1 ) and G 2 = V; E 2 ) such that G 2 is a supergraph of G 1 , a graph G ....

.... has been studied intensely because of its application to sparse matrix algebra [9] It is known to be NP complete even for binary trees [16] and for caterpillars with hair length at most three [34] On the other hand, it is solvable in O(n k ) for arbitrary k [23] and in linear time for k = 2 [16]. Sandwich Problems: Given two graphs G 1 = V; E 1 ) and G 2 = V; E 2 ) such that G 2 is a supergraph of G 1 , a graph G = V; E) is called a sandwich for this pair if E 1 E E 2 . G must be sandwiched between G 1 and G 2 , hence the name) Let E 3 be the set of all ....

[Article contains additional citation context not shown here]

M. R. Garey, R. L. Graham, D. S. Johnson, and D. E. Knuth. Complexity results for bandwidth minimization. SIAM J. Appl. Math., 34(3):477--495, 1978.

....as to minimize the width of this band. When the (symmetric) nonzero elements of the matrix are viewed as vertex adjacencies, then the reordering problem is exactly the minimum bandwidth problem of this graph. The bandwidth problem is known to be NP hard [Pap76] even for certain families of trees [GGJK78, Mon86]. Recent results bound the approximability of the bandwidth problem. Feige [Fei97] gives a nearly linear time randomized algorithm that finds linear arrangement with bandwidth within ratio of O(log 9=2 n) of the optimum. Blache, Karpinski and Wirtgen [BKW98] show that the bandwidth problem on ....

M. R. Garey, R. L. Graham, D. S. Johnson, and D. E. Knuth. Complexity results for bandwidth minimization. SIAM J. Appl. Math., 34(3):477--495, 1978.

....by bw(f ) is defined to be bw(f) maxfjf(v) Gamma f(w)j : vw 2 Eg: The bandwidth of the graph G is bw(G) minfbw(f) f is a layout of Gg: If bw(f) k (or bw(G) k) we speak briefly of a bw k layout (resp. a bw k graph) The paper of Chung [1] is a good survey on graph labelings. In [2] Garey, Graham, Johnson, and Knuth presented a very sophisticated linear time algorithm for recognizing bw 2 graphs. Saxe [5] found, for each fixed k, a relatively easy algorithm which recognizes bw k graphs with complexity O(n k 1 ) i.e. bw 2 graphs with complexity O(n 3 ) Finally ....

M.R. Garey, R.L. Graham, D.S. Johnson, and D.E. Knuth, Complexity results for bandwidth minimization, SIAM J. Appl. Math., 34 (1978), pp. 477--495.

.... with interval graphs, of importance to both algorithmic and structural graph theory, and with applications to VLSI layout [6, 11] Bandwidth is another widely studied graph parameter, with applications to sparse matrices [1] and notorious for the difficulty of its computation even for trees [3]. Recently, in a study of problems motivated by molecular biology, Kaplan and Shamir [5] showed a somewhat surprising connection between bandwidth and pathwidth, using the well known notion of proper interval graphs, or equivalently unit interval graphs. A similar connection was shown by [2] based ....

M. Garey, R. Graham, D. Johnson and D. Knuth, Complexity results for bandwidth minimization, SIAM J. Applied Math, vol. 34. no. 3, May 1978.

....G fB(f; G)g Clearly the bandwidth of G is the greatest bandwidth of its components. 4 2 3 6 5 7 8 1 Figure 1: A small 1=4 dense graph G. It has 8 vertices and minimum degree 2. The problem of finding the bandwidth of a graph is NP complete [Pa 76] even for trees with maximum degree 3 [GGJK 78] The general problem is not known to have any sublinear n ffl approximation algorithms. There are only few cases where we can find the optimal layout in polynomial time. Saxe [Sa 80] designed an algorithm which decides whether a given graph has bandwidth at most k in time O(n k ) by dynamic ....

....n ffl approximation algorithms. There are only few cases where we can find the optimal layout in polynomial time. Saxe [Sa 80] designed an algorithm which decides whether a given graph has bandwidth at most k in time O(n k ) by dynamic programming. Bandwidth two can be checked in linear time [GGJK 78] Kratsch [Kr 87] introduced an exact O(n 2 log n) algorithm for the bandwidth problem in interval graphs. Smithline [Sm 95] proved that the bandwidth of the complete k ary tree T k;d with d levels and k d leaves is exactly dk(k d Gamma 1) k Gamma 1) 2d)e. Her proof is constructive and ....

Garey, M., Graham, R., Johnson, D., Knuth, D., Complexity Results For Bandwidth Minimization, SIAM J. Appl. Math. 34 (1978), pp. 477--495. 10

.... with interval graphs, of importance to both algorithmic and structural graph theory, and with applications to VLSI layout [6, 11] Bandwidth is another widely studied graph parameter, with applications to sparse matrices [1] and notorious for the difficulty of its computation even for trees [3]. Recently, in a study of problems motivated by molecular biology, Kaplan and Shamir [5] showed a somewhat surprising connection between bandwidth and pathwidth, using the well known notion of proper interval graphs, or equivalently unit interval graphs. A similar connection was shown by [2] based ....

M. Garey, R. Graham, D. Johnson and D. Knuth, Complexity results for bandwidth minimization, SIAM J. Applied Math, vol. 34. no. 3, May 1978.

....be inherently sequential. In the next paragraph we explore what is known about the complexity of some problems defined on trees and also explain why finding a P complete problem defined on trees is useful. The bandwidth problem, which does not involve weights, restricted to trees is NP complete ([6], 17] If weights are allowed, there are other natural tree problems that are known to be NP complete (see, for example, 18] A P complete problem, whose definition is based solely on the structure of trees, might prove useful in helping to resolve the complexity of a number of open problems, ....

....0 1 1 0 0 0 0 0 6 0 1 2 3 4 5 6 7 8 In step 6 the arrays LeftClear and RightClear are constructed. The element LeftClear[i] specifies how many values i is greater than before encountering a node of size greater than or equal to i in a left to right traversal of the chain. For example, LeftClear[6] = 2 indicates that before encountering a node of size greater than or equal to 6, in this case 7, 6 is greater than two nodes, 4 and 3, when traversing from the left end of the chain. 0 0 0 0 2 2 2 5 1 2 3 4 5 6 7 8 RightClear has a similar meaning but in the right to left direction. 0 1 2 2 2 ....

M. R. Garey, R. L. Graham, D. S. Johnson, and D. E. Knuth. Complexity results for bandwidth minimization. SIAM Journal on Applied Mathematics, 34:477--495, 1978.

....adjacent vertices in G corresponding to f . The bandwidth B(G) is then B(G) min f is a numbering of G fB(f; G)g Clearly the bandwidth of G is the greatest bandwidth of its components. The problem of finding the bandwidth of a graph is NP hard [Pa 76] even for trees with maximum degree 3 [GGJK 78] The general problem is not known to have any sublinear n ffl approximation algorithms. There are only few cases where we can find the optimal layout in polynomial time. Saxe [Sa 80] designed an algorithm which decides whether a given graph has bandwidth at most k in time O(n k ) by dynamic ....

....n ffl approximation algorithms. There are only few cases where we can find the optimal layout in polynomial time. Saxe [Sa 80] designed an algorithm which decides whether a given graph has bandwidth at most k in time O(n k ) by dynamic programming. Bandwidth two can be checked in linear time [GGJK 78] Kratsch [Kr 87] introduced an exact O(n 2 log n) algorithm for the bandwidth problem in interval graphs. Smithline [Sm 95] proved that the bandwidth of the complete k ary tree T k;d with d levels and k d leaves is exactly dk(k d Gamma 1) k Gamma 1) 2d)e. Her proof is constructive and ....

Garey, M., Graham, R., Johnson, D., Knuth, D., Complexity Results for Bandwidth Minimization, SIAM J. Appl. Math. 34 (1978), pp. 477-- 495.

....of G fB(f; G)g Clearly the bandwidth of G is the greatest bandwidth of its components. 4 2 3 6 5 7 8 1 Figure 1: A small 1=4 dense graph G. It has 8 vertices and minimum degree 2. The problem of finding the bandwidth of a graph is NP complete [Pa 76] even for trees with maximum degree 3 [GGJK 78] The general problem is not known to have any sublinear n ffl approximation algorithms. There are only few cases where we can find the optimal layout in polynomial time. Saxe [Sa 80] designed an algorithm which decides whether a given graph has bandwidth at most k in time O(n k ) by dynamic ....

....n ffl approximation algorithms. There are only few cases where we can find the optimal layout in polynomial time. Saxe [Sa 80] designed an algorithm which decides whether a given graph has bandwidth at most k in time O(n k ) by dynamic programming. Bandwidth two can be checked in linear time [GGJK 78] Kratsch [Kr 87] introduced an exact O(n 2 log n) algorithm for the bandwidth problem in interval graphs. Smithline [Sm 95] proved that the bandwidth of the complete k ary tree T k;d with d levels and k d leaves is exactly dk(k d Gamma 1) k Gamma 1) 2d)e. Her proof is constructive ....

Garey, M., Graham, R., Johnson, D., Knuth, D., Complexity Results For Bandwidth Minimization, SIAM J. Appl. Math. 34 (1978), pp. 477--495.

....f is a numbering of G fB(f; G)g Clearly the bandwidth of G is the greatest bandwidth of its components. Therefore, we assume without loss of generality that the input graph is connected. The problem of constructing the bandwidth of a graph is NP hard [Pa 76] even for trees with maximum degree 3 [GGJK 78] There are only few cases for which we can construct the optimal layout in polynomial time [GGJK 78] Sa 80] Ch 88] Sm 95] To date there was not much known about approximating the bandwidth. Recently Feige [Fe 97] constructed an approximation algorithm constructing a layout within a ....

....Therefore, we assume without loss of generality that the input graph is connected. The problem of constructing the bandwidth of a graph is NP hard [Pa 76] even for trees with maximum degree 3 [GGJK 78] There are only few cases for which we can construct the optimal layout in polynomial time [GGJK 78] Sa 80] Ch 88] Sm 95] To date there was not much known about approximating the bandwidth. Recently Feige [Fe 97] constructed an approximation algorithm constructing a layout within a polylogarithmic factor of the optimum. The algorithm (cf. Fe 97] is based on volume respecting ....

Garey, M., Graham, R., Johnson, D., Knuth, D., Complexity Results For Bandwidth Minimization, SIAM J. Appl. Math. 34 (1978), pp. 477--495.

....adjacent vertices in G corresponding to f . The bandwidth B(G) is then B(G) min f is a numbering of G fB(f; G)g Clearly the bandwidth of G is the greatest bandwidth of its components. The problem of finding the bandwidth of a graph is NP hard [Pa 76] even for trees with maximum degree 3 [GGJK 78] The general problem is not known to have any sublinear n ffl approximation algorithms. There are only few cases where we can find the optimal layout in polynomial time. Saxe [Sa 80] designed an algorithm which decides whether a given graph has bandwidth at most k in time O(n k ) by dynamic ....

....n ffl approximation algorithms. There are only few cases where we can find the optimal layout in polynomial time. Saxe [Sa 80] designed an algorithm which decides whether a given graph has bandwidth at most k in time O(n k ) by dynamic programming. Bandwidth two can be checked in linear time [GGJK 78] Kratsch [Kr 87] introduced an exact O(n 2 log n) algorithm for the bandwidth problem in interval graphs. Smithline [Sm 95] proved that the bandwidth of the complete k ary tree T k;d with d levels and k d leaves is exactly dk(k d Gamma 1) k Gamma 1) 2d)e. Her proof is constructive and ....

Garey, M., Graham, R., Johnson, D., Knuth, D., Complexity Results For Bandwidth Minimization, SIAM J. Appl. Math. 34 (1978), pp. 477--495.

....G(A) has bandwidth b if and only if there is a permutation matrix P such that in PAP t all nonzero entries appear within b of the main diagonal. In this case A has bandwidth b. The problem of constructing the bandwidth of a graph is NP hard [Pa 76] even for trees with maximum degree 3 [GGJK 78] There are only few cases where we can construct the optimal layout in polynomial time. Saxe [Sa 80] designed an algorithm which decides whether a given graph has bandwidth at most k in time O(n k 1 ) by dynamic programming. His algorithm can be turned into a construction algorithm of the ....

....layout in polynomial time. Saxe [Sa 80] designed an algorithm which decides whether a given graph has bandwidth at most k in time O(n k 1 ) by dynamic programming. His algorithm can be turned into a construction algorithm of the optimum layout. Bandwidth two can be checked in linear time [GGJK 78] Smithline [Sm 95] see also [Ch 88] proved that the bandwidth of the complete k ary tree T k;d with d levels and k d leaves is exactly dk(k d Gamma 1) k Gamma 1) 2d)e. The proof is constructive and entails a polynomial time algorithm for this problem. The topological bandwidth is a ....

Garey, M., Graham, R., Johnson, D., Knuth, D., Complexity Results For Bandwidth Minimization, SIAM J. Appl. Math. 34 (1978), pp. 477--495.

....section for the class of AT free claw free graphs. Both problems are NP hard in general. For Bandwidth, this was shown first by Papadimitriou [22] Subsequently, NP hardness even when restricted to trees with maximum degree three or to caterpillars with hairs of length at most three was proved [12, 21]. For the Treewidth problem, NP hardness even for co bipartite graphs was shown by Arnborg et al. 1] Clearly, co bipartite graphs are AT free and clawfree. Thus we obtain from Theorem 12 immediately 2 : Theorem14. The Bandwidth problem is NP hard for co bipartite graphs. Polynomial time ....

M. R. Garey, R. L. Graham, D. S. Johnson, and D. E. Knuth. Complexity results for bandwidth minimization. SIAM J. Appl. Math., 34:477--495, 1978.

....length among paths in H used to represent edges in G. The bandwidth of a graph G is the minimum dilation among injective simulations of G using a path as the host graph. Computing the bandwidth of graphs in general is NP complete [3] Moreover, this is even true of trees with maximum degree 3 [1]. It is thus of interest to consider this problem on Typeset by A M S T E X 2 DENNIS EICHHORN, DHRUV MUBAYI, KEVIN O BRYANT, AND DOUGLAS B. WEST restricted classes of graphs. We study a parameter that corresponds to the restriction of the bandwidth parameter to the class of line graphs. The ....

M. R. Garey, R.L. Graham, D.S. Johnson, D.E. Knuth, Complexity results for bandwidth minimization, SIAM J. Appl. Math. 34 (1978), 477--495.

.... Gamma (v)j. The complexity of a graph in terms of a linear layout is usually obtained by measuring length, crossing edges or neighbors placement. The bandwidth problem asks for a layout minimizing the maximum edge length. The problem is NP complete [28] even for trees with maximum degree 3 [14] or caterpillars with hair length 3 [25] It can be approximated within a constant for some restricted classes of trees [18] but has no polynomial time approximation scheme for trees [4] It has a constant randomized approximation algorithm for dense instances [21] and no polynomial time ....

M. R. Garey, R. L. Graham, D. S. Johnson, and D. Knuth. Complexity results for bandwidth minimization. SIAM J on Applied Mathematics, 34:477--495, September 1978.

....: jVjg. The width b(G; L) of L is zero, if G has no edges, and otherwise b(G; L) maxfjL(u) L(v)j j fu; vg 2 e.g. The bandwidth of G is bw(G) minfb(G; L) j L is a layout of Gg: In general, computing the bandwidth of a graph is NP complete [29] even for trees having maximum degree three [14]. There are only a few graph classes known for which the bandwidth can be computed efficiently. One such graph class is the class of theta graphs [31] Another one is the class of caterpillars with hairs of length one and two [2] However, for caterpillars with hairs of length at most three, the ....

.... class of theta graphs [31] Another one is the class of caterpillars with hairs of length one and two [2] However, for caterpillars with hairs of length at most three, the BANDWIDTH problem remains NP complete [27] It can be checked in linear time whether the bandwidth of a graph is at most two [14]. For general k, there is an O(n k ) algorithm to check whether the bandwidth of a graph is at most k [16] In some sense, this is best possible, since it was shown in [3] that BANDWIDTH is W[t] hard for all t in the fixed parameter hierarchy. Hence it cannot be expected that there is an O(n ....

Garey, M. R., R. L. Graham, D. S. Johnson and D. E. Knuth, Complexity results for bandwidth minimization, SIAM Journal on Applied Mathematics 34 (1978), pp. 477--495.

....numbering of its vertices as above such that for every vertex v its numbering is greater than any numbering of a vertex u such that (u; v) 2 E. The DBANDWIDTH problem corresponds to that of minimizing the bandwidth of an upper triangular matrix by simultaneous row and column permutations (cf. [GGJK78]) The problem is known to be NP hard even if restricted to binary trees (cf. GGJK78] or caterpillars with hairs of length at most 3 [M83] This makes the BANDWIDTH one of the very rare combinatorial problems which are computationally hard for trees. Interestingly, the problem is ....

....than any numbering of a vertex u such that (u; v) 2 E. The DBANDWIDTH problem corresponds to that of minimizing the bandwidth of an upper triangular matrix by simultaneous row and column permutations (cf. GGJK78] The problem is known to be NP hard even if restricted to binary trees (cf. [GGJK78]) or caterpillars with hairs of length at most 3 [M83] This makes the BANDWIDTH one of the very rare combinatorial problems which are computationally hard for trees. Interestingly, the problem is efficiently computable for complete trees [Sm95] Only a very few special cases of this problem ....

M. Garey, R. Graham, D. Johnson, D. Knuth, Complexity Results for Bandwidth Minimization, SIAM J. Appl. Math. 34 (1978), pp. 477-495.

....caterpillars. We also demonstrate that the local density bound need not be optimal for block graphs of diameter 3 or for trees of diameter 4. Fig. 1. Sketch of a block caterpillar Computing bandwidth is NP complete [14] even for trees with maximum degree 3 [4]; hence the interest in special classes. Slightly enlarging the classes of 2 caterpillars or block caterpillars yields classes on which bandwidth is NP complete. Monien [12] proved that bandwidth is NP complete for 3 caterpillars, although he needs paths of length 3 only at one vertex of the ....

M.R. Garey, R.L. Graham, D.S. Johnson, and D.E. Knuth, Complexity results for bandwidth minimization. SIAM J. Appl. Math. 34(1978), 477--495.

....constraint graph can be used as a branching order that reduces backtracking. For a survey on the bandwidth problem and early approaches, see [2] The minimum bandwidth problem was first shown to be NP hard in 1976 [15] and later even for trees of degree at most three and for caterpillars [7, 13]. Approximations algorithms have been known only for some special families of graphs, such as caterpillars or asteroidal triple free graphs [11, 12] 2 The Semi Definite Relaxation Our approximation algorithm begins with an SDP (Semi Definite Programming) relaxation. First we motivate and ....

M Garey, R. Graham, D. Johnson, D. E. Knuth, Complexity results for bandwidth minimization, SIAM J. Appl. Math. 34: 477--495, 1978.

....of G fB(f; G)g Clearly the bandwidth of G is the greatest bandwidth of its components. 4 2 3 6 5 7 8 1 Figure 1: A small 1=4 dense graph G. It has 8 vertices and minimum degree 2. The problem of finding the bandwidth of a graph is NP complete [Pa 76] even for trees with maximum degree 3 [GGJK 78] The general problem is not known to have any sublinear n ffl approximation algorithms. There are only few cases where we can find the optimal layout in polynomial time. Saxe [Sa 80] designed an algorithm which decides whether a given graph has bandwidth at most k in time O(n k ) by ....

....ffl approximation algorithms. There are only few cases where we can find the optimal layout in polynomial time. Saxe [Sa 80] designed an algorithm which decides whether a given graph has bandwidth at most k in time O(n k ) by dynamic programming. Bandwidth two can be checked in linear time [GGJK 78] Kratsch [Kr 87] introduced an exact O(n 2 log n) algorithm for the bandwidth problem in interval graphs. Smithline [Sm 95] proved that the bandwidth of the complete k ary tree T k;d with d levels and k d leaves is exactly dk(k d Gamma 1) k Gamma 1) 2d)e. Her proof is constructive and ....

Garey, M., Graham, R., Johnson, D., Knuth, D., Complexity Results For Bandwidth Minimization, SIAM J. Appl. Math. 34 (1978), pp. 477--495.

....adjacent vertices in G corresponding to f . The bandwidth B(G) is then B(G) min f is a numbering of G fB(f; G)g Clearly the bandwidth of G is the greatest bandwidth of its components. The problem of finding the bandwidth of a graph is NP hard [Pa 76] even for trees with maximum degree 3 [GGJK 78] The general problem is not known to have any sublinear n ffl approximation algorithm (cf. KWZ 97] Ka 97] Smithline [Sm 95] proved that the bandwidth of a complete k ary tree can be computed in polynomial time. For caterpillars [HMM 91] found a polynomial time log n approximation ....

Garey, M., Graham, R., Johnson, D., Knuth, D., Complexity Results For Bandwidth Minimization, SIAM J. Appl. Math. 34 (1978), pp. 477--495.

....now natural to ask the same problem with unit intervals: Here we can obtain an even stronger result, using the following recent theorem from [27] Computing the bandwidth is equivalent to finding a unit interval supergraph with minimum clique size. Hence, the NPcompleteness of BANDWIDTH on trees [17, 40], immediately implies that given a graph G, finding a unit interval supergraph of G whose clique size is minimum is NP complete, even if G is a binary tree, or a caterpillar with hair length three. 5 k Consecutive Ones Matrices Recall that the k consecutive ones property (abbreviated to k C1P) ....

M. R. Garey, R. L. Graham, D. S. Johnson, and D. E. Knuth. Complexity results for bandwidth minimization. SIAM J. Appl. Math., 34(3):477--495, 1978.

....O(logn) approximable [47] O(log log n) approx. for planar graphs [47] PTAS for dense graphs [5] MinSumCut sc( P i=1: n jffi(i; j NP C [25] P for trees [17] NC for trees [17] Bandwidth bw( maxuv2E (uv; NP C [41] APX for certain trees [28] NP C for trees Delta = 3 [24] APX for dense graphs [32] and caterpillars hair length 3 [37] no PTAS for trees [9] no APX in general [31] MinCut cut( maxi=1: n j (i; j NP C [27] P for trees [51] NP C for pl. graphs Delta = 3 [38] NC for trees [16] APX for dense graphs [5] VertSep vs( maxi=1: n ....

M. R. Garey, R. L. Graham, D. S. Johnson, and D. Knuth. Complexity results for bandwidth minimization. SIAM J on Applied Mathematics, 34:477--495, September 1978.

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M. R. Garey, R. L. Graham, D. S. Johnson, and D. E. Knuth, Complexity results for bandwidth minimization, SIAM J. Appl. Math., 34 (1978), pp. 477--495.

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M. Garey, R. Graham, D. Johnson, and D. Knuth. Complexity results for bandwidth minimization. SIAM Journal on Applied Mathematics, 34:477--495, 1978.

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M. Garey, R. Graham, D. Johnson, and D. Knuth. Complexity results for bandwidth minimization. 34:477--495, 1978.

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M.R. Garey, R.L. Graham, D.S. Johnson and D.E. Knuth, \Complexity Results for Bandwidth Minimization", SIAM Journal on Applied Mathematics 34 (1978) 477-495.

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M. R. Garey, R. L. Graham, D. S. Johnson, and D. E. Knuth. Complexity results for bandwidth minimization. SIAM Journal on Applied Mathematics, 34(3):477--495, May 1978.

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M. R. Garey, R. L. Graham, D. S. Johnson, and D. E. Knuth, Complexity results for bandwidth minimization, SIAM J. Appl. Math., 34 (1978), pp. 477--495.

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M. R. Garey, R. L. Graham, D. S. Johnson, and D. E. Knuth, Complexity results for bandwidth minimization, SIAM J. Appl. Math., 34 (1978), pp. 477--495.

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M. R. Garey, R. L. Graham, D. S. Johnson, and D. E. Knuth, Complexity results for bandwidth minimization, SIAM J. Appl. Math., 34 (1978), pp. 477--495.

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