Bandwidth of convex bipartite graphs and related graphs

Shrestha, Anish Man Singh; Tayu, Satoshi; Ueno, S.

doi:10.1016/j.ipl.2012.02.012

Cited by 9 publications

(13 citation statements)

References 13 publications

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“…As AT-free graphs, cocomparability graphs, permutation graphs, trapezoid graphs, convex bipartite graphs, caterpillars with hairs of bounded length, all have bounded path-length or have k-dominating shortest paths with constant k, they admit constant factor approximations of the minimum bandwidth and the minimum line-distortion. Thus, the constant factor approximation results of [26,31,41] become special cases of our results.…”

Section: Discussionsupporting

confidence: 50%

Line-Distortion, Bandwidth and Path-Length of a Graph

Dragan

Köhler

Leitert

2014

Algorithm Theory – SWAT 2014

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Abstract. For a graph G = (V, E) the minimum line-distortion problem asks for the minimum k such that there is a mapping f of the vertices into points of the line such that for each pair of vertices x, y the distance on the line |f (x) − f (y)| can be bounded by the term dG(x, y) ≤ |f (x) − f (y)| ≤ k dG(x, y), where dG(x, y) is the distance in the graph. The minimum bandwidth problem minimizes the term maxuv∈E |f (u) − f (v)|, where f is a mapping of the vertices of G into the integers {1, . . . , n}. We investigate the minimum line-distortion and the minimum bandwidth problems on unweighted graphs and their relations with the minimum length of a Robertson-Seymour's path-decomposition. The length of a path-decomposition of a graph is the largest diameter of a bag in the decomposition. The path-length of a graph is the minimum length over all its path-decompositions. In particular, we show:-if a graph G can be embedded into the line with distortion k, then G admits a Robertson-Seymour's path-decomposition with bags of diameter at most k in G; -for every class of graphs with path-length bounded by a constant, there exist an efficient constantfactor approximation algorithm for the minimum line-distortion problem and an efficient constantfactor approximation algorithm for the minimum bandwidth problem; -there is an efficient 2-approximation algorithm for computing the path-length of an arbitrary graph; -AT-free graphs and some intersection families of graphs have path-length at most 2; -for AT-free graphs, there exist a linear time 8-approximation algorithm for the minimum linedistortion problem and a linear time 4-approximation algorithm for the minimum bandwidth problem.

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Section: Discussionsupporting

confidence: 50%

Line-Distortion, Bandwidth and Path-Length of a Graph

Dragan

Köhler

Leitert

2014

Algorithm Theory – SWAT 2014

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“…Especially, it is N P-complete even if G is restricted to a caterpillar with hair length 3 [13]. The bandwidth problem is N P-complete not only for trees, but also for split graphs [14] and convex bipartite graphs [15]. From the viewpoint of exact algorithms, the problem seems to be a difficult one; Feige developed an O(10 n ) time exact algorithm for the bandwidth problem of general graphs in 2000 [16], and recently, Cygan and Pilipczuk improved it to O(4.383 n ) time (see [17][18][19] for further details).…”

Section: Introductionmentioning

confidence: 99%

“…From the viewpoint of exact algorithms, the problem seems to be a difficult one; Feige developed an O(10 n ) time exact algorithm for the bandwidth problem of general graphs in 2000 [16], and recently, Cygan and Pilipczuk improved it to O(4.383 n ) time (see [17][18][19] for further details). Therefore approximation algorithms for several graph classes have been developed (see, e.g., [15,[20][21][22][23]). Only a few graph classes have been known for which the bandwidth problem can be solved in polynomial time.…”

Section: Introductionmentioning

confidence: 99%

Tractabilities and Intractabilities on Geometric Intersection Graphs

Uehara

2013

Algorithms

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A graph is said to be an intersection graph if there is a set of objects such that each vertex corresponds to an object and two vertices are adjacent if and only if the corresponding objects have a nonempty intersection. There are several natural graph classes that have geometric intersection representations. The geometric representations sometimes help to prove tractability/intractability of problems on graph classes. In this paper, we show some results proved by using geometric representations.

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“…We know that the minimum bandwidth problem is NP-hard even on bounded path-width graphs (e.g., even on caterpillars of hair-length at most 3 [36,13]). Recently, in [41], it was shown that the minimum bandwidth problem is NP-hard also on so-called convex bipartite graphs. A bipartite graph G = (U, V ; E) is said to be convex if for one of its parts, say U , there is an ordering (u 1 , u 2 , .…”

Section: ✷(Claim)mentioning

confidence: 99%

Line-Distortion, Bandwidth and Path-Length of a Graph

2015

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For a graph G = (V, E) the minimum line-distortion problem asks for the minimum k such that there is a mapping f of the vertices into points of the line such that for each pair of vertices x, y the distance on the line |f (x) − f (y)| can be bounded by the term dG(x, y) ≤ |f (x) − f (y)| ≤ k dG(x, y), where dG(x, y) is the distance in the graph. The minimum bandwidth problem minimizes the term maxuv∈E |f (u) − f (v)|, where f is a mapping of the vertices of G into the integers {1, . . . , n}. We investigate the minimum line-distortion and the minimum bandwidth problems on unweighted graphs and their relations with the minimum length of a Robertson-Seymour's path-decomposition. The length of a path-decomposition of a graph is the largest diameter of a bag in the decomposition. The path-length of a graph is the minimum length over all its path-decompositions. In particular, we show:if a graph G can be embedded into the line with distortion k, then G admits a Robertson-Seymour's path-decomposition with bags of diameter at most k in G; -for every class of graphs with path-length bounded by a constant, there exist an efficient constantfactor approximation algorithm for the minimum line-distortion problem and an efficient constantfactor approximation algorithm for the minimum bandwidth problem; -there is an efficient 2-approximation algorithm for computing the path-length of an arbitrary graph; -AT-free graphs and some intersection families of graphs have path-length at most 2; -for AT-free graphs, there exist a linear time 8-approximation algorithm for the minimum linedistortion problem and a linear time 4-approximation algorithm for the minimum bandwidth problem.

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Bandwidth of convex bipartite graphs and related graphs

Cited by 9 publications

References 13 publications

Line-Distortion, Bandwidth and Path-Length of a Graph

Line-Distortion, Bandwidth and Path-Length of a Graph

Tractabilities and Intractabilities on Geometric Intersection Graphs

Line-Distortion, Bandwidth and Path-Length of a Graph

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