Hardness of approximating the undirected minimum size k-spanner

Lecture Notes in Computer Science, 1997

The problem of finding a minimum weight k-vertex connected spanning subgraph is considered. For k ≥ 2, this problem is known to be NP-hard. Combining properties of inclusion-minimal k-vertex connected graphs and of k-out-connected graphs (i.e., graphs which contain a vertex from which there exist k internally vertex-disjoint paths to every other vertex), we derive polynomial time approximation algorithms for the cases k = 3, 4, 5. * Up to 1990, E. A. Dinic, Moscow.

Given a k vertex connected graph with weighted edges, we study the problem of nding a minimum weight spanning subgraph which is k vertex-connected, for k = 2; 3; 4. The problem is known to be NP-hard for any k 2, even when edges have no weight.

Information and Computation, 2013

We present an O(√ n log n)-approximation algorithm for the problem of finding the sparsest spanner of a given directed graph G on n vertices. A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, given a graph G = (V, E) with nonnegative edge lengths d : E → R ≥0 and a stretch k ≥ 1, a subgraph H = (V, E H) is a k-spanner of G if for every edge (s, t) ∈ E, the graph H contains a path from s to t of length at most k • d(s, t). The previous best approximation ratio wasÕ(n 2/3), due to Dinitz and Krauthgamer (STOC '11). We also improve the approximation ratio for the important special case of directed 3-spanners with unit edge lengths fromÕ(√ n) to O(n 1/3 log n). The best previously known algorithms for this problem are due to Berman, Raskhodnikova and Ruan (FSTTCS '10) and Dinitz and Krauthgamer. The approximation ratio of our algorithm almost matches Dinitz and Krauthgamer's lower bound for the integrality gap of a natural linear programming relaxation. Our algorithm directly implies an O(n 1/3 log n)-approximation for the 3-spanner problem on undirected graphs with unit lengths. An easy O(√ n)-approximation algorithm for this problem has been the best known for decades. Finally, we consider the Directed Steiner Forest problem: given a directed graph with edge costs and a collection of ordered vertex pairs, find a minimumcost subgraph that contains a path between every prescribed pair. We obtain an approximation ratio of O(n 2/3+) for any constant > 0, which improves the O(n • min(n 4/5 , m 2/3)) ratio due to Feldman, Kortsarz and Nutov (SODA '09).

SIAM Journal on Discrete Mathematics, 1990

The problem offinding a minimum-weight k-connected spanning subgraph ofa complete graph, assuming that the edge weights satisfy the triangle inequality, is studied. It is shown that the class of minimumweight k-edge connected spanning subgraphs can be restricted to those subgraphs which, in addition to the connectivity requirements, satisfy the following two conditions: (I) Every vertex has degree k or k + 1; (II) Removing any l, 2, ..-, or k edges does not leave the resulting connected components all k-edge connected. For the k-vertex connected case, the parallel result is obtained with "k-edge" replaced by "k-vertex," with the added technical restriction that V >= 2k for condition (I) to hold. This generalizes recent work of Monma, Munson, and Pulleyblank for the case k 2. Key words, survivability, graph theory, lifting, connectivity AMS(MOS) subject classifications. 05C40, 94C 15, 90C35 1. Introduction. In the design of communication or transportation networks, it is frequently important to produce networks of low "cost" which are also "survivable." In many cases the cost arises, to a good degree of approximation, in the form ofedge weights that satisfy the "triangle inequality" (defined in precise form below). The overall cost, or weight, or a network is the sum of the individual edge weights. For survivability reasons, the network must satisfy certain connectivity requirements (see [CW], [GM], [MS], [SWK] for more motivation). A typical survivability requirement is that the removal of any (k or fewer edges (or vertices) leaves the remaining network connected. The following standard definitions are required to make the above statements precise. A graph or network G (V, E) is called k-edge connected if the removal of any (k or fewer edges leaves G connected. If, in addition, the removal of any (k or fewer vertices leaves the remaining vertices of G connected, then G is called k-vertex connected. We note that the degenerate graph consisting of a single vertex is k-edge and k-vertex connected for all values of k. A variation of Menger's Theorem states that a nondegenerate graph G is k-edge (respectively, k-vertex) connected if and only if there are k edge (respectively, vertex) disjoint paths between every pair of vertices in G. Hence we obtain the following problem, k-connected network design with triangle inequality: given a complete graph with edge weights that satisfy the triangle inequality, and an integer k, find a minimum-weight k-edge (or k-vertex) connected spanning subgraph. We remark that for any k >_-2 this problem is NP-Hard, as the Hamiltonian Cycle problem can be reduced to a 2-connected network design problem with triangle inequality. Further, in general there will be a difference between the "edge-connected" and "vertex-connected" versions of this problem. In the following, the word "spanning" will be omitted, for convenience. A solution will be a k-connected subgraph. An optimal subgraph or solution will be a solution of least total weight. This paper presents some strong structural properties that optimal subgraphs can be assumed to satisfy. In particular, our results show that there are optimal subgraphs

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?

Journal of Algorithms, 1999

The problem of finding a minimum weight k-vertex connected spanning subgraph in a graph G = (V, E) is considered. For k ≥ 2, this problem is known to be NP-hard. Combining properties of inclusionminimal k-vertex connected graphs and of k-out-connected graphs (i.e., graphs which contain a vertex from which there exist k internally vertex-disjoint paths to every other vertex), we derive an auxiliary polynomial time algorithm for finding a ( k 2 + 1)-connected subgraph with a weight at most twice the optimum to the original problem. In particular, we obtain a 2-approximation algorithm for the case k = 3 of our problem. This improves the best previously known approximation ratio 3. The complexity of the algorithm is O(|V | 3 |E|) = O(|V | 5 ). * Up to 1990,

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.

Information Processing Letters, 1989

The Steiner problem on networks asks for a shortest subgraph spanning a given subset of distinguished vertices. We give a !-approximation algorithm for the special case in which the underlying network is complete and all edge lengths are either 1 or 2. We also relate the Steiner problem to a complexity class recently defined by Papadimitriou and Yannakakis by showing that this special case is MAX SNP-hard, which may be evidence that the Steiner problem on networks has no polynomial-time approximation scheme.

Lecture Notes in Computer Science, 2007

We show that minimal k-vertex connected spanning subgraphs of a given graph can be generated in incremental polynomial time for any fixed k.

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 .