Results 1  10
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11
An Approximation Scheme for Planar Graph TSP
, 1995
"... We consider the special case of the traveling salesman problem (TSP) in which the distance metric is the shortestpath metric of a planar unweighted graph. We present a polynomialtime approximation scheme (PTAS) for this problem. ..."
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Cited by 48 (7 self)
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We consider the special case of the traveling salesman problem (TSP) in which the distance metric is the shortestpath metric of a planar unweighted graph. We present a polynomialtime approximation scheme (PTAS) for this problem.
On VLSI Layouts Of The Star Graph And Related Networks
, 1994
"... . We prove that the minimal VLSI layout of the arrangement graph A(n; k) occupies \Theta(n!=(n \Gamma k \Gamma 1)!) 2 area. As a special case we obtain an optimal layout for the star graph S n with the area \Theta(n!) 2 : This answers an open problem posed by Akers, Harel and Krishnamurthy [1]. ..."
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Cited by 18 (3 self)
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. We prove that the minimal VLSI layout of the arrangement graph A(n; k) occupies \Theta(n!=(n \Gamma k \Gamma 1)!) 2 area. As a special case we obtain an optimal layout for the star graph S n with the area \Theta(n!) 2 : This answers an open problem posed by Akers, Harel and Krishnamurthy [1]. The method is also applied to the pancake graph. The results provide optimal upper and lower bounds for crossing numbers of the above graphs. Key Words: area, arrangement graph, congestion, crossing number, embedding, layout, pancake graph, star graph, VLSI 1
Crossing Numbers: Bounds and Applications
 I. B'AR'ANY AND K. BOROCZKY, BOLYAI SOCIETY MATHEMATICAL STUDIES 6
, 1997
"... We give a survey of techniques for deriving lower bounds and algorithms for constructing upper bounds for several variations of the crossing number problem. Our aim is to emphasize the more general results or those results which have an algorithmic flavor, including the recent results of the autho ..."
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Cited by 13 (5 self)
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We give a survey of techniques for deriving lower bounds and algorithms for constructing upper bounds for several variations of the crossing number problem. Our aim is to emphasize the more general results or those results which have an algorithmic flavor, including the recent results of the authors. We also show applications of crossing numbers to other areas of discrete mathematics, like discrete geometry.
Nested dissection: A survey and comparison of various nested dissection algorithms
, 1992
"... Methods for solving sparse linear systems of equations can be categorized under two broad classes direct and iterative. Direct methods are methods based on gaussian elimination. This report discusses one such direct method namely Nested dissection. Nested Dissection, originally proposed by Alan Geo ..."
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Cited by 8 (1 self)
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Methods for solving sparse linear systems of equations can be categorized under two broad classes direct and iterative. Direct methods are methods based on gaussian elimination. This report discusses one such direct method namely Nested dissection. Nested Dissection, originally proposed by Alan George, is a technique for solving sparse linear systems efficiently. This report is a survey of some of the work in the area of nested dissection and attempts to put it together using a common framework.
Edge Separators For Graphs Of Bounded Genus With Applications
, 1993
"... We prove that every nvertex graph of genus g and maximal degree k has an edge separator of size O( gkn). The upper bound is best possible to within a constant factor. This extends known results on planar graphs and similar results about vertex separators. We apply the edge separator to the is ..."
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Cited by 7 (1 self)
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We prove that every nvertex graph of genus g and maximal degree k has an edge separator of size O( gkn). The upper bound is best possible to within a constant factor. This extends known results on planar graphs and similar results about vertex separators. We apply the edge separator to the isoperimetric number problem, graph embeddings and lower bounds for crossing numbers.
Partitioning Graphs with Costs and Weights on Vertices: Algorithms and Applications
 of Lecture Notes in Computer Science
"... We prove separator theorems in which the size of the separator is minimized with respect to nonnegative vertex costs. We show that for any planar graph G there exists a vertex separator of total vertex cost at most c qP v2V (G) (cost(v)) 2 and that this bound is optimal within a constant factor ..."
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Cited by 4 (0 self)
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We prove separator theorems in which the size of the separator is minimized with respect to nonnegative vertex costs. We show that for any planar graph G there exists a vertex separator of total vertex cost at most c qP v2V (G) (cost(v)) 2 and that this bound is optimal within a constant factor. Moreover such a separator can be found in linear time. This theorem implies a variety of other separation results discussed in the paper. We describe application of our separator theorems to graph embedding problems, graph pebbling, and multi commodity flow problems. 1 Introduction Background. A separator is a small set of vertices or edges whose removal divides a graph into two roughly equal parts. The existence of small separators for some important classes of graphs such as planar graphs can be used in the design of efficient divideandconquer algorithms for problems on such graphs. Formally, a separator theorem for a given class of graphs S states that any nvertex graph from S ca...
On the Minimum Load Coloring Problem
"... Given a graph G = (V,E) with n vertices, m edges and maximum vertex degree ∆, the load distribution of a coloring ϕ: V → {red, blue} is a pair dϕ = (rϕ,bϕ), where rϕ is the number of edges with at least one endvertex colored red and bϕ is the number of edges with at least one endvertex colored blu ..."
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Cited by 1 (0 self)
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Given a graph G = (V,E) with n vertices, m edges and maximum vertex degree ∆, the load distribution of a coloring ϕ: V → {red, blue} is a pair dϕ = (rϕ,bϕ), where rϕ is the number of edges with at least one endvertex colored red and bϕ is the number of edges with at least one endvertex colored blue. Our aim is to find a coloring ϕ such that the (maximum) load, lϕ: = 1 m · max{rϕ,bϕ}, is minimized. This problems arises in Wavelength Division Multiplexing (WDM), the technology currently in use for building optical communication networks. After proving that the general problem is NPhard we give a polynomial time algorithm for optimal colorings of trees and show that the optimal load is at most 1/2+(∆/m)log 2 n. For graphs with genus g> 0, we show that a coloring with load OPT(1 + o(1)) can be computed in O(n + g log n)time, if the maximum degree satisfies ∆ = o ( m2) and an embedding is given. In the general situation we show that a coloring with load at most 3 4 +O(�∆/m) can be found by analyzing a random coloring with Chebychev’s inequality. This bound describes the “typical ” situation: in the random graph model G(n,m) we prove that for almost all graphs, the optimal load is at least 3 4 −�n/m. Finally, we state some conjectures on how our results generalize to k–colorings for k> 2. Key words: graph coloring, graph partitioning
A Bipartite Strengthening of the Crossing Lemma
"... Abstract. The celebrated Crossing Lemma states that, in every drawing of a graph with n vertices and m ≥ 4n edges there are at least Ω(m 3 /n 2) pairs of crossing edges; or equivalently, there is an edge that crosses Ω(m 2 /n 2) other edges. We strengthen the Crossing Lemma for drawings in which any ..."
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Cited by 1 (1 self)
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Abstract. The celebrated Crossing Lemma states that, in every drawing of a graph with n vertices and m ≥ 4n edges there are at least Ω(m 3 /n 2) pairs of crossing edges; or equivalently, there is an edge that crosses Ω(m 2 /n 2) other edges. We strengthen the Crossing Lemma for drawings in which any two edges cross in at most O(1) points. We prove for every k ∈ N that every graph G with n vertices and m ≥ 3n edges drawn in the plane such that any two edges intersect in at most k points has two disjoint subsets of edges, E1 and E2, each of size at least ckm 2 /n 2, such that every edge in E1 crosses all edges in E2, where ck> 0 only depends on k. This bound is best possible up to the constant ck for every k ∈ N. We also prove that every graph G with n vertices and m ≥ 3n edges drawn in the plane with xmonotone edges has disjoint subsets of edges, E1 and E2, each of size Ω(m 2 /(n 2 polylog n)), such that every edge in E1 crosses all edges in E2. On the other hand, we construct xmonotone drawings of bipartite dense graphs where the largest such subsets E1 and E2 have size O(m 2 /(n 2 log(m/n))). 1
Grids
, 2011
"... We shall prove theorems of the following flavor (see textbook/papers for precise statements and proofs). Thm. For any planar graph G = (V, E) on n = V  vertices and for any 1 weight function w: V → R +, we can partition V into A, B, S ⊆ V such that • [α–balanced] w(A), w(B) ≤ α · w(V) for some α ..."
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We shall prove theorems of the following flavor (see textbook/papers for precise statements and proofs). Thm. For any planar graph G = (V, E) on n = V  vertices and for any 1 weight function w: V → R +, we can partition V into A, B, S ⊆ V such that • [α–balanced] w(A), w(B) ≤ α · w(V) for some α ∈ (0, 1) • [separation] no edge between any a ∈ A and b ∈ B (A × B ∩ E = ∅) • [small separator] S  ≤ f(n) • [efficient] A, B, S can be found in linear time. Trees what if G is a (binary) tree? can do 1/2–balanced partition with S  = 1? � only 2/3–balanced! with one edge in separator? � only 3/4–balanced for binary trees