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19
On the Hardness of Approximating Spanners
 Algorithmica
, 1999
"... A k\Gammaspanner of a connected graph G = (V; E) is a subgraph G 0 consisting of all the vertices of V and a subset of the edges, with the additional property that the distance between any two vertices in G 0 is larger than the distance in G by no more than a factor of k. This paper concerns ..."
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Cited by 49 (11 self)
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A k\Gammaspanner of a connected graph G = (V; E) is a subgraph G 0 consisting of all the vertices of V and a subset of the edges, with the additional property that the distance between any two vertices in G 0 is larger than the distance in G by no more than a factor of k. This paper concerns the hardness of finding spanners with a number of edges close to the optimum. It is proved that for every fixed k, approximating the spanner problem is at least as hard as approximating the set cover problem We also consider a weighted version of the spanner problem, and prove an essential difference between the approximability of the case k = 2, and the case k 5. Department of Computer Science, The Open University, 16 Klauzner st., Ramat Aviv, Israel, guyk@shaked.openu.ac.il. 1 Introduction The concept of graph spanners has been studied in several recent papers in the context of communication networks, distributed computing, robotics and computational geometry [ADDJ90, C94, CK94,...
Distance Approximating Trees for Chordal and Dually Chordal Graphs
, 1999
"... In this paper we show that, for each chordal graph G, there is a tree T such that T is a spanning tree of the square G² of G and, for every two vertices, the distance between them in T is not larger than the distance in G plus 2. Moreover, we prove that, if G is a strongly chordal graph or even a ..."
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Cited by 31 (16 self)
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In this paper we show that, for each chordal graph G, there is a tree T such that T is a spanning tree of the square G² of G and, for every two vertices, the distance between them in T is not larger than the distance in G plus 2. Moreover, we prove that, if G is a strongly chordal graph or even a dually chordal graph, then there exists a spanning tree T of G that is an additive 3spanner as well as a multiplicative 4spanner of G. In all cases the tree T can be computed in linear time
Treedecompositions with bags of small diameter
, 2007
"... This paper deals with the length of a Robertson–Seymour’s treedecomposition. The treelength of a graph is the largest distance between two vertices of a bag of a treedecomposition, minimized over all treedecompositions of the graph. The study of this invariant may be interesting in its own right ..."
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Cited by 14 (1 self)
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This paper deals with the length of a Robertson–Seymour’s treedecomposition. The treelength of a graph is the largest distance between two vertices of a bag of a treedecomposition, minimized over all treedecompositions of the graph. The study of this invariant may be interesting in its own right because the class of bounded treelength graphs includes (but is not reduced to) bounded chordality graphs (like interval graphs, permutation graphs, ATfree graphs, etc.). For instance, we show that the treelength of any outerplanar graph is ⌈k/3⌉, where k is the chordality of the graph, and we compute the treelength of meshes. More fundamentally we show that any algorithm computing a treedecomposition approximating the treewidth (or the treelength) of an nvertex graph by a factor α or less does not give an αapproximation of the treelength (resp. the treewidth) unless if α = Ω(n 1/5). We complete these results presenting several polynomial time constant approximate algorithms for the treelength. The introduction of this parameter is motivated by the design of compact distance labeling, compact routing tables with nearoptimal route length, and by the construction of sparse additive spanners.
Additive spanners for kchordal graphs
 In 5 th Italian Conference on Algorithms and Complexity (CIAC), volume 2653 of LNCS
, 2003
"... Abstract. In this paper we show that every chordal graph with n vertices and m edges admits an additive 4spanner with at most 2n−2 edges and an additive 3spanner with at most O(n · log n) edges. This significantly improves results of Peleg and Schäffer from [Graph Spanners, J. Graph Theory, 13(198 ..."
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Cited by 11 (4 self)
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Abstract. In this paper we show that every chordal graph with n vertices and m edges admits an additive 4spanner with at most 2n−2 edges and an additive 3spanner with at most O(n · log n) edges. This significantly improves results of Peleg and Schäffer from [Graph Spanners, J. Graph Theory, 13(1989), 99116]. Our spanners are additive and easier to construct. An additive 4spanner can be constructed in linear time while an additive 3spanner is constructable in O(m · log n) time. Furthermore, our method can be extended to graphs with largest induced cycles of length k. Any such graph admits an additive (k + 1)spanner with at most 2n − 2 edges which is constructable in O(n · k + m) time. Classification: Algorithms, Sparse Graph Spanners 1
Computing geometric minimumdilation graphs is NPhard
 In Proc. of Graph Drawing
, 2006
"... Abstract. Consider a geometric graph G, drawn with straight lines in the plane. For every pair a, b of vertices of G, we compare the shortestpath distance between a and b in G (with Euclidean edge lengths) to their actual Euclidean distance in the plane. The worstcase ratio of these two values, for ..."
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Cited by 9 (0 self)
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Abstract. Consider a geometric graph G, drawn with straight lines in the plane. For every pair a, b of vertices of G, we compare the shortestpath distance between a and b in G (with Euclidean edge lengths) to their actual Euclidean distance in the plane. The worstcase ratio of these two values, for all pairs of vertices, is called the vertextovertex dilation of G. We prove that computing a minimumdilation graph that connects a given npoint set in the plane, using not more than a given number m of edges, is an NPhard problem, no matter if edge crossings are allowed or forbidden. In addition, we show that the minimum dilation tree over a given point set may in fact contain edge crossings.
On spanners of geometric graphs
 In Proc. of the 10th Scandinavian Workshop on Algorithm Theory (SWAT), volume 4059 of LNCS
, 2006
"... Given a connected geometric graph G, we consider the problem of constructing a tspanner of G having the minimum number of edges. We prove that for every t with 1 < t < 1 log n, there exists a con4 nected geometric graph G with n vertices, such that every tspanner of G contains Ω(n1+1/t) edges. Th ..."
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Cited by 5 (2 self)
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Given a connected geometric graph G, we consider the problem of constructing a tspanner of G having the minimum number of edges. We prove that for every t with 1 < t < 1 log n, there exists a con4 nected geometric graph G with n vertices, such that every tspanner of G contains Ω(n1+1/t) edges. This bound almost matches the known upper bound, which states that every connected weighted graph with n vertices contains a tspanner with O(tn1+2/(t+1) ) edges. We also prove that the problem of deciding whether a given geometric graph contains a tspanner with at most K edges is NPhard. Previously, this NPhardness result was only known for nongeometric graphs. 1
A PTAS for the Sparsest Spanners Problem on ApexMinorFree Graphs
, 2008
"... A tspanner of a graph G is a spanning subgraph S in which the distance between every pair of vertices is at most t times their distance in G. The sparsest tspanner problem asks to find, for a given graph G andanintegert, atspanner of G with the minimum number of edges. On general nvertex graph ..."
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Cited by 5 (0 self)
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A tspanner of a graph G is a spanning subgraph S in which the distance between every pair of vertices is at most t times their distance in G. The sparsest tspanner problem asks to find, for a given graph G andanintegert, atspanner of G with the minimum number of edges. On general nvertex graphs, the problem is known to be NPhard for all t ≥ 2, and, even more, it is NPhard to approximate it with ratio O(log n) for every t ≥ 2. For t ≥ 5, the problem remains NPhard for planar graphs, and up to now the approximability status of the problem on planar graphs considered to be open. In this note, we resolve this open issue by showing that the sparsest tspanner problem admits a polynomial time approximation scheme (PTAS) for every t ≥ 1. Actually, our results hold for a much wider class of graphs, namely, on the class of apexminorfree graphs which contains the classes of planar and bounded genus graphs.
Approximation Algorithms for Finding Sparse 2Spanners of 4Connected Planar Triangulations
, 1999
"... A tspanner of an undirected, unweighted graph G is a spanning subgraph S with the added property that for every pair of vertices in G, the distance between them in S is at most t times the distance between them in G. We are interested in finding a sparsest tspanner. In the general setting, this pr ..."
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Cited by 4 (2 self)
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A tspanner of an undirected, unweighted graph G is a spanning subgraph S with the added property that for every pair of vertices in G, the distance between them in S is at most t times the distance between them in G. We are interested in finding a sparsest tspanner. In the general setting, this problem is known to be NPComplete for all t 2. For t 5, the problem remains NPComplete for planar graphs, whereas for t 2 f2; 3; 4g the complexity of this problem on planar graphs is still unknown. In this paper we present approximation algorithms to approximate a sparsest 2spanner of a 4connected planar triangulation as a step towards solving the complexity of this problem in the more general planar setting. 1 Introduction A tspanner of an undirected, unweighted graph G is a spanning subgraph S with the added property that for every pair of vertices in G, the distance between them in S is at most t times the distance between them in G. We are interested in finding a sparsest tspann...
Sparse Additive Spanners for Bounded TreeLength Graphs
 In 11th International Colloquium on Structural Information & Communication Complexity (SIROCCO
, 2004
"... This paper concerns construction of additive stretched spanners with few edges for nvertex graphs having a treedecomposition into bags of diameter at most , i.e., the treelength graphs. For such graphs we construct additive 2spanners with O(n log n) edges, and additive 6spanners with O(n) e ..."
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Cited by 4 (1 self)
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This paper concerns construction of additive stretched spanners with few edges for nvertex graphs having a treedecomposition into bags of diameter at most , i.e., the treelength graphs. For such graphs we construct additive 2spanners with O(n log n) edges, and additive 6spanners with O(n) edges. We also show a lower bound, and prove that there are graphs of treelength for which every multiplicative spanner (and thus every additive ( 1)spanner) requires 1+1=() ) edges.
Spanners for bounded treelength graphs
, 2007
"... This paper concerns construction of additive stretched spanners with few edges for nvertex graphs having a treedecomposition into bags of diameter at most δ, i.e., the treelength δ graphs. For such graphs we construct additive 2δspanners with O(δn+n log n) edges, and additive 4δspanners with O( ..."
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Cited by 3 (2 self)
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This paper concerns construction of additive stretched spanners with few edges for nvertex graphs having a treedecomposition into bags of diameter at most δ, i.e., the treelength δ graphs. For such graphs we construct additive 2δspanners with O(δn+n log n) edges, and additive 4δspanners with O(δn) edges. This provides new upper bounds for chordal graphs for which δ = 1. We also show a lower bound, and prove that there are graphs of treelength δ for which every multiplicative δspanner (and thus every additive (δ − 1)spanner) requires Ω(n 1+1/Θ(δ) ) edges.