K-minimum Spanning Tree Problems Research Articles

Hybrid Approaches based on Simulated Annealing, Tabu Search and Ant Colony Optimization for Solving the k-Minimum Spanning Tree Problem

In graph theory, the k-minimum spanning tree problem is considered to be one of the well-known NP hard problems to solve. This paper address this problem by proposing several hybrid approximate approaches based on the combination of simulated annealing, tabu search and ant colony optimization algorithms. The performances of the proposed methods are compared to other approaches from the literature using the same well-known library of benchmark instances.

A 4-approximation algorithm for k-prize collecting Steiner tree problems

This paper studies a 4-approximation algorithm for k-prize collecting Steiner tree problems. This problem generalizes both k-minimum spanning tree problems and prize collecting Steiner tree problems. Our proposed algorithm employs two 2-approximation algorithms for k-minimum spanning tree problems and prize collecting Steiner tree problems. Also our algorithm framework can be applied to a special case of k-prize collecting traveling salesman problems.

A 5-approximation algorithm for the k-prize-collecting Steiner tree problem

In this paper, we consider the k-prize-collecting Steiner tree problem (k-PCST), extending both the prize-collecting Steiner tree problem (PCST) and the k-minimum spanning tree problem (k-MST). In this problem, we are given a connected graph \(G = (V, E)\), a root vertex r and an integer k. Every edge in E has a nonnegative cost. Every vertex in V has a nonnegative penalty cost. We want to find an r-rooted tree F that spans at least k vertices such that the total cost, including the edge cost of the tree F and the penalty cost of the vertices not spanned by F, is minimized. Our main contribution is to present a 5-approximation algorithm for the k-PCST via the methods of primal–dual and Lagrangean relaxation.

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.

A note on the k-minimum spanning tree problem on circles

Ravi et al. [2] provide a polynomial-time exact solution to the problem of finding a minimum-cost spanning tree with k vertices on the boundary of a convex region and a faster algorithm for the special case of all vertices lying on a circle. In this paper we show that the latter algorithm generally does not solve the kMST problem on circles. A new algorithm is presented that solves the problem in a slightly more general way and with a better complexity.

A hybrid algorithm based on tabu search and ant colony optimization for k-minimum spanning tree problems

This paper considers an efficient approximate algorithm for solving k-minimum spanning tree problems which is one of the combinatorial optimization in networks. A new hybrid algorithm based on tabu search and ant colony optimization is provided. Results of numerical experiments show that the proposed method updates some of the best known values with very short time and that the proposed method provides a better performance with solution accuracy over existing algorithms.

An approximate solution method based on tabu search for k-minimum spanning tree problems

This paper considers a new tabu search-based approximate solution algorithm for k-minimum spanning tree problems. One of the features of the proposed algorithm is that it efficiently obtains local optimal solutions without applying minimum spanning tree algorithms. Numerical experimental results show that the proposed method provides a good performance especially for dense graphs in terms of solution accuracy over existing algorithms.

A taub search algorthm for k-minimum spanning tree problems

A taub search algorthm for k-minimum spanning tree problems