The Complexity of Finding Minimal Spanning Subgraphs

Theory and Application of Graphs, 2016

Cover Page Footnote We greatly appreciate the valuable suggestions made by anonymous referees that resulted in an improved paper.

International Journal of Foundations of Computer Science, 2014

We present algorithms that construct a sparse spanning subgraph of a 3-edge-connected graph that preserves 3-edge connectivity or of a 3-vertex-connected graph that preserves 3-vertex connectivity. Our algorithms are conceptually simple and run in O(|E|) time. These simple algorithms can be used to improve the efficiency of the best-known algorithms for 3-edge and 3-vertex connectivity and their related problems, by preprocessing the input graph so as to trim it down to a sparse graph. Afterwards, the original algorithms run in O(|V |) instead of O(|E|) time. Our algorithms generate an adjacency-lists structure to represent the sparse spanning subgraph, so that when a depth-first search is performed over the subgraph based on this adjacency-lists structure it actually traverses the paths in an ear-decomposition of the subgraph. This is useful because many of the existing algorithms for 3-edge-or 3-vertex connectivity and their related problems are based on an ear-decomposition of the given graph. Using such an adjacency-lists structure to represent the input graph would greatly improve the run-time efficiency of these algorithms.

Lecture Notes in Computer Science, 1997

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.

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,

Discrete Applied Mathematics, 2009

We study the complexity of the problem of deciding the existence of a spanning subgraph of a given graph, and of that of finding a maximum (weight) such subgraph. We establish some general relations between these problems, and we use these relations to obtain new NPcompleteness results for maximum (weight) spanning subgraph problems from analogous results for existence problems and from results in extremal graph theory. On the positive side, we provide a decomposition method for the maximum (weight) spanning chordal subgraph problem that can be used, e.g., to obtain a linear (or O(n log n)) time algorithm for such problems in graphs with vertex degree bounded by 3.

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

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.

Discrete Mathematics, 2008

In this paper we present an algorithm to generate all minimal 3-vertex connected spanning subgraphs of an undirected graph with n vertices and m edges in incremental polynomial time, i.e., for every K we can generate K (or all) minimal 3-vertex connected spanning subgraphs of a given graph in O(K 2 log(K)m 2 + K 2 m 3 ) time, where n and m are the number of vertices and edges of the input graph, respectively. This is an improvement over what was previously available and is the same as the best known running time for generating 2-vertex connected spanning subgraphs. Our result is obtained by applying the decomposition theory of 2-vertex connected graphs to the graphs obtained from minimal 3-vertex connected graphs by removing a single edge.

We show that k-vertex connected spanning subgraphs of a given graph can be generated in incremental polynomial time for any fixed k. We also show that generating k-edge connected spanning subgraphs, where k is part of the input, can be done in incremental polynomial time. These results are based on properties of minimally k-connected graphs which might be of independent interest.

Journal of Algorithms, 1999

The problem of nding 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. Based on the paper of Auletta, Dinitz, Nutov and Parente in this issue, we derive a 3-approximation algorithm for k 2 f4; 5g. This improves the best previously known approximation ratios 4 1 6 and 4 17 30 , respectively. The complexity of the suggested algorithm is O(jV j 5 ) for the deterministic and O(jV j 4 log jV j)

1992

Networks, 2004

We give quasipolynomial-time approximation algorithms for designing networks with a minimum degree. Using our methods, one can design networks whose connectivity is specified by "proper" functions, a class of 0 -1 functions indicating the number of edges crossing each cut. We also provide quasipolynomial-time approximation algorithms for finding two-edge-connected spanning subgraphs of approximately minimum degree of a given two-edge-connected graph, and a spanning tree (branching) of approximately minimum degree of a directed graph. The degree of the output network in all cases is guaranteed to be at most (1 ؉ ⑀) times the optimal degree, plus an additive O(log 1؉⑀ n) for any ⑀ > 0. Our analysis indicates that the degree of an optimal subgraph for each of the problems above is well estimated by certain polynomially solvable linear programs. This suggests that the linear programs we describe could be useful in obtaining optimal solutions via branch and bound.

Algorithmica, 1995

The essential subgraph H of a weighted graph or digraph G contains an edge (v, w) if that edge is uniquely the least-cost path between its vertices. Let s denote the number of edges of H. This paper presents an algorithm for solving all-pairs shortest paths on G that requires O(ns + n 2 log n) worst-case running time. In general the time is equivalent to that of solving n single-source problems using only edges in H. For general models of random graphs and digraphs G, s = O(n log n) almost surely. The subgraph H is optimal in the sense that it is the smallest subgraph sufficient for solving shortest-path problems in G. Lower bounds on the largest-cost edge of H and on the diameter of H and G are obtained for general randomly weighted graphs. Experimental results produce some new conjectures about essential subgraphs and distances in graphs with uniform edge costs.

11

2008

The problem of finding a spanning tree with few leaves is motivated by the design of communication networks, where the cost of the devices depends on their routing functionality (ending, forwarding, or routing a connection). Besides this application, the problem has its own theoretical importance as a generalization of the Hamiltonian path problem. Lu and Ravi showed that there is no constant factor approximation for minimizing the number of leaves of a spanning tree, unless P = NP. Thus instead of minimizing the number of leaves, we are going to deal with maximizing the number of non-leaves: we give a linear-time 2-approximation for arbitrary graphs, a 3/2-approximation for claw-free graphs, and a 6/5-approximation for cubic graphs.