Spanning Subgraph Problem Research Articles

A new approximation algorithm for the minimum 2-edge-connected spanning subgraph problem

We present a new approximation algorithm for the minimum 2-edge-connected spanning subgraph problem. Its approximation ratio is 43, which matches the current best ratio. The approximation ratio of the algorithm is 65 on subcubic graphs, which is an improvement upon the previous best ratio of 54. The algorithm is a novel extension of the primal-dual schema, which consists of two distinct phases. Both the algorithm and the analysis are much simpler than those of the previous approaches.

Solving the degree-concentrated fault-tolerant spanning subgraph problem by DC programming

In this paper, we consider the maximum and minimum versions of degree-concentrated fault-tolerant spanning subgraph problem which has many applications in network communications. We prove that both this two problems are NP-hard. For the maximum version, we use DC programming relaxation to propose a heuristic algorithm. Numerical tests indicate that the proposed algorithm is efficient and effective. For the minimum version, we also formulate it as a DC program, and show that the DC algorithm does not perform well for this problem.

Two-Connected Spanning Subgraphs with at Most $\frac{10}{7}{OPT}$ Edges

We present a $\frac{10}{7}$-approximation algorithm for the minimum 2-vertex-connected spanning subgraph problem. Similarly to the work of Cheriyan, Sebö, and Szigeti for 2-edge-connected spanning subgraphs, our algorithm is based on computing a carefully designed ear-decomposition.

On the minimum-cost $$\lambda $$ λ -edge-connected k-subgraph problem

In this paper, we propose several integer programming (IP) formulations to exactly solve the minimum-cost $$\lambda $$ -edge-connected k-subgraph problem, or the $$(k,\lambda )$$ -subgraph problem, based on its graph properties. Special cases of this problem include the well-known k-minimum spanning tree problem (if $$\lambda =1$$ ), $$\lambda $$ -edge-connected spanning subgraph problem (if $$k=|V|$$ ) and k-clique problem (if $$\lambda = k-1$$ and there are exact k vertices in the subgraph). As a generalization of k-minimum spanning tree and a case of the $$(k,\lambda )$$ -subgraph problem, the (k, 2)-subgraph problem is studied, and some special graph properties are proved to find stronger and more compact IP formulations. Additionally, we study the valid inequalities for these IP formulations. Numerical experiments are performed to compare proposed IP formulations and inequalities.

The Partial Inverse Minimum Connected Spanning Subgraph Problem with Given Cyclomatic Number

In this paper, we consider a partial inverse problem of minimum connected spanning subgraph with cyclomatic number . That is, given a subgraph, a cyclomatic number and a constraint that the edge weights can only decrease, we want to modify the edge weights as little as possible, so that there exists a minimum connected spanning subgraph with cyclomatic number and containing the given subgraph. For the case that the given subgraph is a connected subgraph with cyclomatic number , we solve the problem in polynomial time by contracting the subgraph. And when the given subgraph is a spanning tree, we solve the problem in polynomial time by using the MST algorithm.

Power optimization for connectivity problems

Given a graph with costs on the edges, the power of a node is the maximum cost of an edge leaving it, and the power of the graph is the sum of the powers of the nodes of this graph. Motivated by applications in wireless multi-hop networks, we consider four fundamental problems under the power minimization criteria: the Min-Power b-Edge-Cover problem (MPb-EC) where the goal is to find a min-power subgraph so that the degree of every node v is at least some given integer b(v), the Min-Power k-node Connected Spanning Subgraph problem (MPk-CSS), Min-Power k-edge Connected Spanning Subgraph problem (MPk-CSS), and finally the Min-Power k-Edge-Disjoint Paths problem in directed graphs (MPk-EDP). We give an O(log4 n)-approximation algorithm for MPb-EC. This gives an O(log4 n)-approximation algorithm for MPk-CSS for most values of k, improving the best previously known O(k)-approximation guarantee. In contrast, we obtain an $$O(\sqrt{n})$$ approximation algorithm for ECSS, and for its variant in directed graphs (i.e., MPk-EDP), we establish the following inapproximability threshold: MPk-EDP cannot be approximated within O(2log1-e n) for any fixed e > 0, unless NP-hard problems can be solved in quasi-polynomial time.

Graph homomorphisms through random walks

AbstractIn this paper we introduce some general necessary conditions for the existence of graph homomorphisms, which hold in both directed and undirected cases. Our method is a combination of Diaconis and Saloff–Coste comparison technique for Markov chains and a generalization of Haemers interlacing theorem. As some applications, we obtain a necessary condition for the spanning subgraph problem, which also provides a generalization of a theorem of Mohar (1992) as a necessary condition for Hamiltonicity. In particular, in the case that the range is a Cayley graph or an edge‐transitive graph, we obtain theorems with a corollary about the existence of homomorphisms to cycles. This, specially, provides a proof of the fact that the Coxeter graph is a core. Also, we obtain some information about the cores of vertex‐transitive graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 15–38, 2003

A linear time -approximation for the minimum strongly-connected spanning subgraph problem

A linear time ▪-approximation algorithm is presented for the NP-hard problem of finding a minimum strongly-connected spanning subgraph. It is based on cycle contraction that was first introduced by Khuller, Raghavachari and Young [SIAM J. Comput. 24 (1995) 859–872]. We improve their result by contracting special cycles and utilizing a more efficient data structure.

A Better Approximation Ratio for the Minimum Sizek-Edge-Connected Spanning Subgraph Problem

Consider the minimum sizek-edge-connected spanning subgraph problem: given a positive integerkand ak-edge-connected graphG, find ak-edge-connected spanning subgraph ofGwith the minimum number of edges. This problem is known to be NP-complete. Khuller and Raghavachari presented the first algorithm which, for allk, achieves a performance ratio smaller than a constant which is less than two. They proved an upper bound of 1.85 for the performance ratio of their algorithm. Currently, the best known performance ratio for the problem is 1+2/(k+1), achieved by a slower algorithm of Cheriyan and Thurimella. In this article, we improve Khuller and Raghavachari's analysis, proving that the performance ratio of their algorithm is smaller than 1.7 for large enoughk, and that it is at most 1.75 for allk. Second, we show that the minimum size 2-edge-connected spanning subgraph problem is MAX SNP-hard.

A linear time algorithm for the Bottleneck Biconnected Spanning Subgraph problem

A linear time algorithm for the Bottleneck Biconnected Spanning Subgraph problem is presented. This improves the hitherto best-known solution, which has a running time of O( m + n log n ), where m and n are the number of edges and vertices of the graph.