Restrictions of Minimum Spanner Problems
udi rotics1997, Information and Computation
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Abstract
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.
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Approximation of minimum weight spanners for sparse graphs
Theoretical Computer Science, 2011
A t-spanner of a graph G is its spanning subgraph S such that the distance between every pair of vertices in S is at most t times their distance in G. The sparsest t-spanner problem asks to find, for a given graph G and an integer t, a t-spanner of G with the minimum number of edges. The problem is known to be NP-hard for all t ≥ 2, and, even more, it is NP-hard to approximate it with ratio O(log n) for every t ≥ 2. For t ≥ 5, the problem remains NP-hard for planar graphs and the approximability status of the problem on planar graphs was open. We resolve this open issue by showing that the sparsest t-spanner problem admits the efficient polynomial time approximation scheme (EPTAS) for every t ≥ 1. Our result holds for a much wider class of graphs, namely, the class of apex-minor-free graphs, which contains the classes of planar and bounded genus graphs. Moreover, it is possible to extend our results to weighted apex-minor free graphs, when the maximum edge weight is bounded by some constant.
Spanners in sparse graphs
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?
A PTAS for the Sparsest Spanners Problem on Apex-Minor-Free Graphs
Lecture Notes in Computer Science, 2008
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. Graphs of bounded treewidth are sparse graphs and our technique can be used to settle the complexity of the parameterized version of the sparse tspanner 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 sparse t-spanner problem is fixed parameter tractable on apexminor-free graphs with t and m being the parameters. Finally we show that the tractability border of the t-spanner problem cannot be extended beyond the class of apex-minor-free graphs. In particular, we prove that for every t ≥ 4, the problem of finding a tree t-spanner is NP-complete on K6-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.
A simple and linear time randomized algorithm for computing sparse spanners in weighted graphs
Random Structures and Algorithms, 2007
Let G = (V, E) be an undirected weighted graph on |V | = n vertices, and |E| = m edges. A t-spanner of the graph G, for any t ≥ 1, is a subgraph (V, E S ), E S ⊆ E, such that the distance between any pair of vertices in the subgraph is at most t times the distance between them in the graph G. Computing a t-spanner of minimum size (number of edges) has been a widely studied and well motivated problem in computer science. In this paper we present the first linear time randomized algorithm that computes a t-spanner of a given weighted graph. Moreover, the size of the t-spanner computed essentially matches the worst case lower bound implied by a 43 years old girth conjecture made independently by Erdős [26], Bollobás [19], and Bondy & Simonovits .
Tree Spanners for Bipartite Graphs and Probe Interval Graphs
Algorithmica, 2007
A tree t-spanner T in a graph G is a spanning tree of G such that the distance between every pair of vertices in T is at most t times their distance in G. The tree t-spanner problem asks whether a graph admits a tree t-spanner, given t. We first substantially strengthen the known results for bipartite graphs. We prove that the tree t-spanner problem is NP-complete even for chordal bipartite graphs for t ≥ 5, and every bipartite ATE-free graph has a tree 3-spanner, which can be found in linear time. The best known before results were NP-completeness for general bipartite graphs, and that every convex graph has a tree 3-spanner. We next focus on the tree t-spanner problem for probe interval graphs and related graph classes. The graph classes were introduced to deal with the physical mapping of DNA. From a graph theoretical point of view, the classes are natural generalizations of interval graphs. We show that these classes are tree 7-spanner admissible, and a tree 7-spanner can be constructed in O(m log n) time.
On Additive Spanners in Weighted Graphs with Local Error
Graph-Theoretic Concepts in Computer Science, 2021
An additive +β spanner of a graph G is a subgraph which preserves distances up to an additive +β error. Additive spanners are well-studied in unweighted graphs but have only recently received attention in weighted graphs [Elkin et al. 2019 and 2020, Ahmed et al. 2020]. This paper makes two new contributions to the theory of weighted additive spanners. For weighted graphs, [Ahmed et al. 2020] provided constructions of sparse spanners with global error β = cW , where W is the maximum edge weight in G and c is constant. We improve these to local error by giving spanners with additive error +cW (s, t) for each vertex pair (s, t), where W (s, t) is the maximum edge weight along the shortest s-t path in G. These include pairwise +(2 + ε)W (•, •) and +(6 + ε)W (•, •) spanners over vertex pairs P ⊆ V × V on Oε(n|P| 1/3) and Oε(n|P| 1/4) edges for all ε > 0, which extend previously known unweighted results up to ε dependence, as well as an all-pairs +4W (•, •) spanner on O(n 7/5) edges. Besides sparsity, another natural way to measure the quality of a spanner in weighted graphs is by its lightness, defined as the total edge weight of the spanner divided by the weight of an MST of G. We provide a +εW (•, •) spanner with Oε(n) lightness, and a +(4 + ε)W (•, •) spanner with Oε(n 2/3) lightness. These are the first known additive spanners with nontrivial lightness guarantees. All of the above spanners can be constructed in polynomial time.
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A 4/3-Approximation Algorithm for the Minimum 2-Edge Connected Subgraph Problem
ACM Transactions on Algorithms, 2019
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Collective additive tree spanners of bounded tree-breadth graphs with generalizations and consequences
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).
The Complexity of Finding Minimal Spanning Subgraphs
1991
Let P be a property of graphs (directed or undirected). We consider the following problem: given a graph G that has property P , find a minimal spanning subgraph of G with property P . We describe an algorithm for this problem and prove that it is correct under some rather weak assumptions about P . We then analyze the number of iterations of this algorithm. By suitably restricting the graph properties, we devise a general technique to construct graphs for which the algorithm requires a large number of iterations. We apply the above technique to three concrete graph properties: 2-edge-connectivity, biconnectivity, and strong connectivity. We obtain a tight lower bound of\Omega\Gamma/45 n) on the number of iterations of the algorithm for finding minimal spanning subgraphs with these properties; this resolves open questions posed earlier with regard to these properties. This also implies that the worst case sequential running time of the algorithm for these three properties is \Omega\...
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Theoretical Computer Science, 2004
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A tree t-spanner of a graph G is a spanning subtree 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 the tree stretch index. The problem of determining the tree stretch index has been studied by: establishing lower and upper bounds, based, for instance, on the girth value and on the minimum diameter spanning tree problem, respectively; and presenting some classes for which t is a tight value. Moreover, in 1995, the computational complexities of determining whether \(t = 2\) or \(t \ge 4\) were settled to be polynomially time solvable and NP-complete, respectively, while deciding if \(t = 3\) still remains an open problem.
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In the pairwise weighted spanner problem, the input consists of a weighted directed graph on n vertices, where each edge is assigned both a cost and a length. Furthermore, we are given k terminal vertex pairs and a distance constraint for each pair. The goal is to find a minimum-cost subgraph in which the distance constraints are satisfied. A more restricted variant of this problem was shown to be O(2 log 1−ε n)-hard to approximate under a standard complexity assumption, by Elkin and Peleg (Theory of Computing Systems, 2007). This general formulation captures many well-studied network connectivity problems, including spanners, distance preservers, and Steiner forests. We study the weighted spanner problem, in which the edges have positive integral lengths of magnitudes that are polynomial in n, while the costs are arbitrary non-negative rational numbers. Our results include the following in the classical offline setting: • AnÕ(n 4/5+ε)-approximation algorithm for the pairwise weighted spanner problem. When the edges have unit costs and lengths, the best previous algorithm gives anÕ(n 3/5+ε)-approximation, due to Chlamtáč, Dinitz, Kortsarz, and Laekhanukit (Transactions on Algorithms, 2020). • AnÕ(n 1/2+ε)-approximation algorithm for the weighted spanner problem when the terminal pairs consist of all vertex pairs and the distances must be preserved exactly. When the edges have unit costs and arbitrary positive lengths, the best previous algorithm gives anÕ(n 1/2)-approximation for the all-pair spanner problem, due to Berman, Bhattacharyya, Makarychev, Raskhodnikova, and Yaroslavtsev (Information and Computation, 2013). We also prove the first results for the weighted spanners in the online setting. In the online setting, the terminal vertex pairs arrive one at a time, in an online fashion, and edges are required to be added irrevocably to the solution in order to satisfy the distance constraints, while approximately minimizing the cost. Our results include the following: • AnÕ(k 1/2+ε)-competitive algorithm for the online pairwise weighted spanner problem. The stateof-the-art results are anÕ(n 4/5)-competitive algorithm when edges have unit costs and arbitrary positive lengths, and a min{Õ(k 1/2+ε),Õ(n 2/3+ε)}-competitive algorithm when edges have unit costs and lengths, due to Grigorescu, Lin, and Quanrud (APPROX, 2021).
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