Results 1  10
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27
Approximating the Bandwidth Via Volume Respecting Embeddings
, 1999
"... A linear arrangement of an nvertex graph is a onetoone mapping of its vertices to the integers f1; : : : ; ng. The bandwidth of a linear arrangement is the maximum difference between mapped values of adjacent vertices. The problem of finding a linear arrangement with smallest possible bandwidt ..."
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Cited by 93 (3 self)
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A linear arrangement of an nvertex graph is a onetoone mapping of its vertices to the integers f1; : : : ; ng. The bandwidth of a linear arrangement is the maximum difference between mapped values of adjacent vertices. The problem of finding a linear arrangement with smallest possible bandwidth in NPhard. We present a randomized algorithm that runs in nearly linear time and outputs a linear arrangement whose bandwidth is within a polylogarithmic multiplicative factor of optimal. Our algorithm is based on a new notion, called volume respecting embeddings, which is a natural extension of small distortion embeddings of Bourgain and of Linial, London and Rabinovich. 1 Introduction We consider the problem of minimizing the bandwidth of an undirected connected graph G(V; E), where n = jV j and m = jEj. One needs to find a linear arrangement of the vertices, namely, a onetoone mapping f : V \Gamma! f1; 2; : : : ng, for which the bandwidth, i.e. max (i;j)2E jf(i) \Gamma f(j)j, i...
Treewidth and Minimum Fillin: Grouping the Minimal Separators
, 1999
"... We use the notion of potential maximal clique to characterize the maximal cliques appearing in minimal triangulations of a graph. We show that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum fillin are polynomially tractable for these graphs ..."
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Cited by 31 (5 self)
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We use the notion of potential maximal clique to characterize the maximal cliques appearing in minimal triangulations of a graph. We show that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum fillin are polynomially tractable for these graphs. We prove that for all classes of graphs for which polynomial algorithms computing the treewidth and the minimum fillin exist, we can list their potential maximal cliques in polynomial time. Our approach unies these algorithms. Finally we show how to compute in polynomial time the potential maximal cliques of weakly triangulated graphs, for which the treewidth and the minimum fillin problems were open.
SemiDefinite Relaxations for Minimum Bandwidth and other VertexOrdering problems
 THEOR. COMPUT. SCI
, 2000
"... We present simple semidefinite programming relaxations for the NPhard minimum bandwidth and minimum length linear ordering problems. We then show how these relaxations can be rounded in a natural way (via random projection) to obtain approximation guarantees for both of these vertexordering pr ..."
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Cited by 27 (4 self)
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We present simple semidefinite programming relaxations for the NPhard minimum bandwidth and minimum length linear ordering problems. We then show how these relaxations can be rounded in a natural way (via random projection) to obtain approximation guarantees for both of these vertexordering problems.
A WideRange Efficient Algorithm For Minimal Triangulation
 Proceedings of SODA'99
, 1999
"... Traditionally, efficient algorithms for computing a minimal triangulation of a graph (i.e. embedding a graph into a triangulated graph by adding an inclusionminimal set of edges) required first computing a special ordering on the vertices of the graph, called a minimal ordering. We give a new algor ..."
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Cited by 20 (9 self)
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Traditionally, efficient algorithms for computing a minimal triangulation of a graph (i.e. embedding a graph into a triangulated graph by adding an inclusionminimal set of edges) required first computing a special ordering on the vertices of the graph, called a minimal ordering. We give a new algorithm which efficiently computes a minimal triangulation using an arbitrary ordering on the vertices. 1 Introduction. Computing a minimal triangulation consists in embedding a given graph into a triangulated graph by adding a set of edges (called a fill). If the set of edges added is inclusionminimal, the fill is said to be minimal, and the corresponding triangulated graph is called a minimal triangulation. Finding a fill that is minimum is NPcomplete ([10]). Given a graph G and any ordering ff on its vertices, an associated fill can be computed by repeatedly choosing the next vertex x in order ff, adding the edges necessary to make the neighborhood of x into a clique (i.e. by making x si...
Additive Tree Spanners
 SIAM JOURNAL ON DISCRETE MATHEMATICS
, 1998
"... A spanning tree of a graph is a kadditive tree spanner whenever the distance of every two vertices in the graph or in the tree differs by at most k. In this paper we show that certain classes of graphs, as distancehereditary graphs, interval graphs, asteroidaltriple free graphs, allow some consta ..."
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Cited by 14 (0 self)
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A spanning tree of a graph is a kadditive tree spanner whenever the distance of every two vertices in the graph or in the tree differs by at most k. In this paper we show that certain classes of graphs, as distancehereditary graphs, interval graphs, asteroidaltriple free graphs, allow some constant k such that every member of the class has some kadditive tree spanner. On the other hand, there are chordal graphs without kadditive tree spanner for arbitrary large k.
Collective tree spanners and routing in ATfree related graphs (Extended Abstract)
 IN GRAPHTHEORETIC CONCEPTS IN COMPUTER SCIENCE, LECTURE NOTES IN COMPUT. SCI. 3353
, 2004
"... In this paper we study collective additive tree spanners for families of graphs that either contain or are contained in ATfree graphs. We say that a graph G = (V, E) admits a system of µ collective additive tree rspanners if there is a system T (G) of at most µ spanning trees of G such that for any ..."
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Cited by 9 (8 self)
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In this paper we study collective additive tree spanners for families of graphs that either contain or are contained in ATfree graphs. We say that a graph G = (V, E) admits a system of µ collective additive tree rspanners if there is a system T (G) of at most µ spanning trees of G such that for any two vertices x, y of G a spanning tree T ∈ T (G) exists such that dT(x, y) ≤ dG(x, y) + r. Among other results, we show that ATfree graphs have a system of two collective additive tree 2spanners (whereas there are trapezoid graphs that do not admit any additive tree 2spanner). Furthermore, based on this collection, we derive a compact and efficient routing scheme. Also, any DSPgraph (there exists a dominating shortest path) admits an additive tree 4spanner, a system of two collective additive tree 3spanners and a system of five collective additive tree 2spanners.
A Note on Linear Discrepancy and Bandwidth
 J. Combin. Math. Combin. Comput
, 2002
"... Fishburn, Tanenbaum and Trenk [4] de ne the linear discrepancy ld(P ) of a poset P = (V; < P ) as the minimum integer k 0 for which there exists a bijection f : V ! f1; 2; : : : ; jV jg such that u < P v implies f(u) < f(v) and ujj P v implies jf(u) f(v)j k. In [5] they prove that the linear disc ..."
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Cited by 7 (0 self)
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Fishburn, Tanenbaum and Trenk [4] de ne the linear discrepancy ld(P ) of a poset P = (V; < P ) as the minimum integer k 0 for which there exists a bijection f : V ! f1; 2; : : : ; jV jg such that u < P v implies f(u) < f(v) and ujj P v implies jf(u) f(v)j k. In [5] they prove that the linear discrepancy of a poset equals the bandwidth of its cocomparability graph.
An Exponential Time 2Approximation Algorithm for Bandwidth
"... The bandwidth of a graph G on n vertices is the minimum b such that the vertices of G can be labeled from 1 to n such that the labels of every pair of adjacent vertices differ by at most b. In this paper, we present a 2approximation algorithm for the Bandwidth problem that takes worstcase O(1.9797 ..."
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Cited by 6 (0 self)
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The bandwidth of a graph G on n vertices is the minimum b such that the vertices of G can be labeled from 1 to n such that the labels of every pair of adjacent vertices differ by at most b. In this paper, we present a 2approximation algorithm for the Bandwidth problem that takes worstcase O(1.9797 n) = O(3 0.6217n) time and uses polynomial space. This improves both the previous best 2 and 3approximation algorithms of Cygan et al. which have an O ∗ (3 n) and O ∗ (2 n) worstcase time bounds, respectively. Our algorithm is based on constructing bucket decompositions of the input graph. A bucket decomposition partitions the vertex set of a graph into ordered sets (called buckets) of (almost) equal sizes such that all edges are either incident on vertices in the same bucket or on vertices in two consecutive buckets. The idea is to find the smallest bucket size for which there exists a bucket decomposition. The algorithm uses a simple divideandconquer strategy along with dynamic programming to achieve this improved time bound. 1