M. Karpinski, J. Wirtgen, and A. Zelikovsky. An approximating algorithm for the bandwidth problem on dense graphs. Technical Report TR 97-017, ECCC, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....general graphs [12, 33] For some particular kinds of graphs, there are di#erent approximations algorithms. Recall that a graph G with n vertices is called # dense i# the minimum degree of G is at least #n. For # dense graphs there exists a 3 approximation in polynomial time, moreover it is in RNC [61]. For an introduction to parallel approximability, see [29] There are also polynomial time O(log n) approximation algorithms for caterpillars [47] and for a more large class of trees, denoted as GHB trees, which are characterized as trees such that for any node v, the depth di#erence of any two ....

M. Karpinski, J. Wirtgen, and A. Zelikovsky. An approximating algorithm for the bandwidth problem on dense graphs. Technical Report TR 97-017, ECCC, 1997.

.... 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 approximation algorithm for general graphs [20] The minimum cut arrangement asks for a layout minimizing the maximum cut along the nested sequence of vertex sets. The problem is NP complete [17] even for planar graphs with maximum degree 3 [26] For trees the problem is ....

M. Karpinski, J. Wirtgen, and A. Zelikovsky. An approximating algorithm for the bandwidth problem on dense graphs. Technical Report TR 97-017, ECCC, 1997. 9

.... and dense graphs, though in the case of random graphs with constant expected degree, we must ensure the existence of a giant component in the random graph [2, 11] We remark, though, that for dense graphs ( E = #( V 2 ) polynomial approximation schemes are known for some of the problems [6, 8]. 2 Approximation results We introduce now a class of graphs that captures the properties we need to bound the gaps for our layout problems on random graphs. Definition 2 (Mixing graphs) Let # # (0, 1 6 ) # # (0, 1) and define C #,# = 3(1 ln 3) ##) 2 . Consider a sequence (c n ) n#1 ....

M. Karpinski, J. Wirtgen, and A. Zelikovsky. An approximating algorithm for the bandwidth problem on dense graphs. Technical Report TR 97-017, ECCC, 1997.

....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 for trees (ref. in [15] P ....

M. Karpinski, J. Wirtgen, and A. Zelikovsky. An approximating algorithm for the bandwidth problem on dense graphs. Technical Report TR 97-017, ECCC, 1997.

.... 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 approximation algorithm for general graphs [20] The minimum cut arrangement asks for a layout minimizing the maximum cut along the nested sequence of vertex sets. The problem is NP complete [17] even for planar graphs with maximum degree 3 [26] For trees the problem is ....

M. Karpinski, J. Wirtgen, and A. Zelikovsky. An approximating algorithm for the bandwidth problem on dense graphs. Technical Report TR 97-017, ECCC, 1997.

....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 jffi(i; j NP C [35] P for ....

M. Karpinski, J. Wirtgen, and A. Zelikovsky. An approximating algorithm for the bandwidth problem on dense graphs. Technical Report TR 97-017, ECCC, 1997.

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M. Karpinski, J. Wirtgen, and A. Zelikovsky. An approximating algorithm for the bandwidth problem on dense graphs. Technical Report TR 97-017, ECCC, 1997.

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