The Complexity of Finding Minimal Spanning Subgraphs (original) (raw)

On Finding Minimal Two-Connected Subgraphs

1992

We present efficient parallel algorithms for the problems of finding a minimal2-edge-connected spanning subgraph of a 2-edge-connected graph and finding a minimal biconnected spanning subgraph of a biconnected graph. The parallel algorithms run in polylog time using a linear number of PRAM processors. We also give linear time sequential algorithms for minimally augmenting a spanning tree into a 2-edge-connected or biconnected graph. 1 Introduction In this paper we consider the following two related problems: given a 2-edge-connected (biconnected) graph G, compute a minimal 2-edge-connected (biconnected) spanning subgraph of G, i.e., a 2-edge-connected (biconnected) subgraph in which the deletion of any edge destroys 2-edgeconnectivity (biconnectivity). We present efficient parallel algorithms for these problems. It is known that the corresponding problems of finding minimum spanning subgraphs with these properties are NP-hard ([6]). Thus, it is natural to study the simpler problem o...

An Experimental Study of Polynomial Time Algorithms for Minimum 2-Edge Connected Subgraph Problem

International Journal of Computer Science and Mobile Computing (IJCSMC), 2024

In this paper, we focus on the problem of finding a minimum-sized directed 2-edge-connected subgraph, a problem classified as NP-Complete [8], which plays a critical role in various practical applications. We present approximation algorithms aimed at finding efficient, high-quality solutions within polynomial time. These algorithms are based on a comprehensive analysis of the problem of finding a directed 2-edge-connected subgraph, with performance evaluated in terms of the number of edges. The results of our experiments demonstrate that the proposed algorithms effectively reduce the number of remaining edges across different graph scenarios, particularly in high-density graphs. Moreover, they maintain strong connectivity even in the event of edge failures, ensuring the continuity of network operations in the face of faults and disasters.

ON FINDING SPARSE THREE-EDGE-CONNECTED AND THREE-VERTEX-CONNECTED SPANNING SUBGRAPHS

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.

Better algorithms for minimum weight vertex-connectivity problems

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.

A 2-Approximation Algorithm for Finding an Optimum 3-Vertex-Connected Spanning Subgraph

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,

On the complexity of some subgraph problems

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.