Optimal Spanners in Partial k-Trees

Journal of Computer and System Sciences, 2011

A t-spanner 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. If S is required to be a tree then S is called a tree t-spanner of G. In 1998, Fekete and Kremer showed that on unweighted planar graphs the Tree t-Spanner problem (the problem to decide whether G admits a tree t-spanner) is polynomial time solvable for t ≤ 3 and is NP-complete as long as t is part of the input. They also left as an open problem whether the Tree t-Spanner problem is polynomial time solvable for every fixed t ≥ 4. In this work we resolve this open problem and extend the solution in several directions. We show that for every fixed t, it is possible in polynomial time not only to decide if a planar graph G has a tree t-spanner, but also to decide if G has a t-spanner of bounded treewidth. Moreover, for every fixed values of t and k, the problem, for a given planar graph G to decide if G has a t-spanner of treewidth at most k, is not only polynomial time solvable, but is fixed parameter tractable (with k and t being the parameters). In particular, the running time of our algorithm is linear with respect to the size of G. We extend this result from planar to a much more general class of sparse graphs containing graphs of bounded genus. An apex graph is a graph obtained from a planar graph G by adding a vertex and making it adjacent to some vertices of G. We show that the problem of finding a t-spanner of treewidth k is fixed parameter tractable on graphs that do not contain some fixed apex graph as a minor, i.e. on apex-minor-free graphs. We prove that the tractability border of the t-spanner problem cannot be extended beyond the class of apex-minor-free graphs. In particular, for every t ≥ 4, the problem of finding a tree t-spanner is NP-complete on K 6 -minor-free graphs. Thus our results are tight, in a sense that the restriction of input graph being apex-minor-free cannot be replaced by H-minor-free for some non-apex fixed graph H. Graphs of bounded treewidth are sparse graphs and our technique can be used to settle the complexity of the parameterized version of the Sparsest t-Spanner problem, where for given t and m one asks if a given n-vertex graph has a t-spanner with at most n − 1 + m edges. Our results imply that the Sparsest t-Spanner problem is fixed parameter tractable on apex-minor-free graphs with t and m being the parameters. Finally, we show that the optimization version of the Sparsest t-Spanner problem, which asks for a t-spanner with the minimum number of edges, admits PTAS for apexminor-free graphs. This resolves an open question asked by Duckworth, Wormald, and Zito. * A preliminary version of these results appeared in the proceedings of the 35th International Colloquium PROBLEM: k-Treewidth t-spanner INSTANCE: A connected graph G and integers k and t. QUESTION: Is there a t-spanner S of G of treewidth at most k?

SIAM Journal on Computing, 2009

We address the following problem: Given a complete k-partite geometric graph K whose vertex set is a set of n points in R d , compute a spanner of K that has a "small" stretch factor and "few" edges. We present two algorithms for this problem. The first algorithm computes a (5 + )-spanner of K with O(n) edges in O(n log n) time. The second algorithm computes a (3 + )-spanner of K with O(n log n) edges in O(n log n) time. The latter result is optimal: We show that for any 2 ≤ k ≤ n − Θ( √ n log n), spanners with O(n log n) edges and stretch factor less than 3 do not exist for all complete k-partite geometric graphs.

Foundations of Information Technology in the Era of Network and Mobile Computing, 2002

A spanning tree T of a graph G is said to be a tree t-spanner if the distance between any two vertices in T is at most t times their distance in G. A graph that has a tree t-spanner is called a tree t-spanner admissible graph. The complexity of recognizing tree 3-spanner admissible graphs is still unknown." In this paper, a characterization of tree 3-spanner admissible 2-trees is presented. Linear time algorithms for recognizing tree 3-spanner admissible 2-trees and for constructing tree 3-spanners in such 2-trees are also proposed.

Theoretical Computer Science, 2014

In this paper, we study collective additive tree spanners for families of graphs enjoying special Robertson-Seymour's tree-decompositions, and demonstrate interesting consequences of obtained results. We say that a graph G admits a system of µ collective additive tree r-spanners (resp., multiplicative tree t-spanners) 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 (resp., dT (x, y) ≤ t · dG(x, y)). When µ = 1 one gets the notion of additive tree r-spanner (resp., multiplicative tree t-spanner). It is known that if a graph G has a multiplicative tree t-spanner, then G admits a Robertson-Seymour's tree-decomposition with bags of radius at most ⌈t/2⌉ in G. We use this to demonstrate that there is a polynomial time algorithm that, given an n-vertex graph G admitting a multiplicative tree t-spanner, constructs a system of at most log 2 n collective additive tree O(t log n)-spanners of G. That is, with a slight increase in the number of trees and in the stretch, one can "turn" a multiplicative tree spanner into a small set of collective additive tree spanners. We extend this result by showing that if a graph G admits a multiplicative t-spanner with tree-width k − 1, then G admits a Robertson-Seymour's tree-decomposition each bag of which can be covered with at most k disks of G of radius at most ⌈t/2⌉ each. This is used to demonstrate that, for every fixed k, there is a polynomial time algorithm that, given an n-vertex graph G admitting a multiplicative t-spanner with tree-width k − 1, constructs a system of at most k(1 + log 2 n) collective additive tree O(t log n)-spanners of G. a stretch t [17], and an additive tree r-spanner of G is a spanning tree with a surplus r [59]. If we approximate the graph by a tree spanner, we can solve the problem on the tree and the solution interpret on the original graph. The tree t-spanner problem asks, given a graph G and a positive number t, whether G admits a tree t-spanner. Note that the problem of finding a tree t-spanner of G minimizing t is known in literature also as the Minimum Max-Stretch spanning Tree problem (see, e.g., and literature cited therein).

In the undirected unweighted minimum size k-spanner problem we are given a graph with edges of cost and length 1, and a number k. The goal is to find a minimum size E and a graph G (V, E) so that for every u, v ∈ V :

Lecture Notes in Computer Science, 2003

In this paper we show that every chordal graph with n vertices and m edges admits an additive 4-spanner with at most 2n−2 edges and an additive 3-spanner with at most O(n · log n) edges. This significantly improves results of Peleg and Schäffer from [Graph Spanners, J. Graph Theory, 13(1989), 99-116]. Our spanners are additive and easier to construct. An additive 4-spanner can be constructed in linear time while an additive 3-spanner 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.

Information and Computation, 1997

A t-spanner of a graph G is a spanning subgraph H such that the distance between any two vertices in H is at most t times their distance in G. Spanners arise in the context of approximating the original graph by a sparse subgraph 23]. The MINIMUM t-SPANNER problem seeks to nd a t-spanner with the minimum number of edges for the given graph. In this paper, we completely settle the complexity status of this problem for various values of t, on Chordal graphs, Split graphs, Bipartite graphs and Convex Bipartite graphs. Our results settle an open question raised in 7] and also greatly simplify some of the proofs presented in 7, 8]. We also give a factor two approximation algorithm for the MINIMUM 2-SPANNER problem on interval graphs. Finally, we provide approximation algorithms for the bandwidth minimization problem on Convex Bipartite graphs and Split graphs using the notion of tree spanners.

SIAM Journal on Discrete Mathematics, 2006

In this paper we introduce a new notion of collective tree spanners. We say that a graph G = (V, E) admits a system of µ collective additive tree r-spanners 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 any chordal graph, chordal bipartite graph or cocomparability graph admits a system of at most log 2 n collective additive tree 2-spanners and any c-chordal graph admits a system of at most log 2 n collective additive tree (2 c/2)-spanners. Towards establishing these results, we present a general property for graphs, called (α, r)decomposition, and show that any (α, r)-decomposable graph G with n vertices admits a system of at most log 1/α n collective additive tree 2rspanners. We discuss also an application of the collective tree spanners to the problem of designing compact and efficient routing schemes in graphs.

Lecture Notes in Computer Science, 2008

In this paper we investigate the (collective) tree spanners problem in homogeneously orderable graphs. This class of graphs was introduced by A. Brandstädt et al. to generalize the dually chordal graphs and the distance-hereditary graphs and to show that the Steiner tree problem can still be solved in polynomial time on this more general class of graphs. In this paper, we demonstrate that every n-vertex homogeneously orderable graph G admits a spanning tree T such that, for any two vertices x, y of G, dT (x, y) ≤ dG(x, y) + 3 (i.e., an additive tree 3-spanner) and a system T (G) of at most O(log n) spanning trees such that, for any two vertices x, y of G, a spanning tree T ∈ T (G) exists with dT (x, y) ≤ dG(x, y) + 2 (i.e, a system of at most O(log n) collective additive tree 2-spanners). These results generalize known results on tree spanners of dually chordal graphs and of distance-hereditary graphs. The results above are also complemented with some lower bounds which say that on some n-vertex homogeneously orderable graphs any system of collective additive tree 1-spanners must have at least Ω(n) spanning trees and there is no system of collective additive tree 2-spanners with constant number of trees.

Cornell University - arXiv, 2022

A tree t-spanner of a graph G is a spanning tree T in which the distance between any two adjacent vertices of G is at most t. The smallest t for which G has a tree t-spanner is called tree stretch index. The t-admissibility problem aims to decide whether the tree stretch index is at most t. Regarding its optimization version, the smallest t for which G is t-admissible is the stretch index of G, denoted by σ T (G). Given a graph with n vertices and m edges, the recognition of 2-admissible graphs can be done O(n + m) time, whereas t-admissibility is NP-complete for σ T (G) ≤ t, t ≥ 4 and deciding if t = 3 is an open problem, for more than 20 years. Since the structural knowledge of classes can be determinant to classify 3-admissibility's complexity, in this paper we present simpler and faster algorithms to check 2 and 3-admissibility for families of graphs with few P 4 's and (k,)-graphs. Regarding (0,)graphs, we present lower and upper bounds for the stretch index of these graphs and characterize graphs whose stretch indexes are equal to the proposed upper bound. Moreover, we prove that t-admissibility is NP-complete even for line graphs of subdivided graphs.