Implications, conflicts, and reductions for Steiner trees

The Steiner tree problem in graphs (SPG) is one of the most studied problems in combinatorial optimization. In the past 10 years, there have been significant advances concerning approximation and complexity of the SPG. However, the state of the art in (practical) exact solution of the SPG has remained largely unchallenged for almost 20 years. While the DIMACS Challenge 2014 and the PACE Challenge 2018 brought renewed interest into Steiner tree problems, even the best new SPG solvers cannot match the state of the art on the vast majority of benchmark instances. The following article seeks to advance exact SPG solution once again. The article is based on a combination of three concepts: Implications, conflicts, and reductions. As a result, various new SPG techniques are conceived. Notably, several of the resulting techniques are (provably) stronger than well-known methods from the literature that are used in exact SPG algorithms. Finally, by integrating the new methods into a branch-and-cut framework, we obtain an exact SPG solver that is not only competitive with, but even outperforms the current state of the art on an extensive collection of benchmark sets. Furthermore, we can solve several instances for the first time to optimality.

The Power of Many: A Physarum Swarm Steiner Tree Algorithm

This paper presents a novel explore-and-fuse approach to solving a large array of problems that cannot be solved by traditional divide-and-conquer. This approach is inspired by Physarum, a unicellular slime mold capable of solving the traveling salesman and Steiner tree problems. Besides exhibiting individual intelligence, Physarum can also share information with other Physarum organisms through fusion. Inspired by the characteristics of Physarum, we spawn many Physarum organisms to explore the problem space in parallel, each gathering information and forming partial solutions pertaining to a local region of the problem space. When the organisms meet, they fuse and share information, eventually forming one organism which has a global view of the problem and can apply its intelligence to find an overall solution to the problem. We demonstrate this novel approach on the NP-hard Steiner tree problem, developing the Physarum Steiner Algorithm. This algorithm is of particular interest due to its ability to leverage parallel processing, avoid obstacles, and operate on various shapes and topological surfaces including the rectilinear grid.

The Clustered Selected-Internal Steiner Tree Problem

Given a complete graph [Formula: see text], with nonnegative edge costs, two subsets [Formula: see text] and [Formula: see text], a partition [Formula: see text] of [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] of [Formula: see text], [Formula: see text], a clustered Steiner tree is a tree [Formula: see text] of [Formula: see text] that spans all vertices in [Formula: see text] such that [Formula: see text] can be cut into [Formula: see text] subtrees [Formula: see text] by removing [Formula: see text] edges and each subtree [Formula: see text] spans all vertices in [Formula: see text], [Formula: see text]. The cost of a clustered Steiner tree is defined to be the sum of the costs of all its edges. A clustered selected-internal Steiner tree of [Formula: see text] is a clustered Steiner tree for [Formula: see text] if all vertices in [Formula: see text] are internal vertices of [Formula: see text]. The clustered selected-internal Steiner tree problem is concerned with the determination of a clustered selected-internal Steiner tree [Formula: see text] for [Formula: see text] and [Formula: see text] in [Formula: see text] with minimum cost. In this paper, we present the first known approximation algorithm with performance ratio [Formula: see text] for the clustered selected-internal Steiner tree problem, where [Formula: see text] is the best-known performance ratio for the Steiner tree problem.

Quasi-Polynomial Algorithms for Submodular Tree Orienteering and Directed Network Design Problems

We consider the following general network design problem. The input is an asymmetric metric (V, c), root [Formula: see text], monotone submodular function [Formula: see text], and budget B. The goal is to find an r-rooted arborescence T of cost at most B that maximizes f(T). Our main result is a simple quasi-polynomial time [Formula: see text]-approximation algorithm for this problem, in which [Formula: see text] is the number of vertices in an optimal solution. As a consequence, we obtain an [Formula: see text]-approximation algorithm for directed (polymatroid) Steiner tree in quasi-polynomial time. We also extend our main result to a setting with additional length bounds at vertices, which leads to improved [Formula: see text]-approximation algorithms for the single-source buy-at-bulk and priority Steiner tree problems. For the usual directed Steiner tree problem, our result matches the best previous approximation ratio but improves significantly on the running time. For polymatroid Steiner tree and single-source buy-at-bulk, our result improves prior approximation ratios by a logarithmic factor. For directed priority Steiner tree, our result seems to be the first nontrivial approximation ratio. Under certain complexity assumptions, our approximation ratios are the best possible (up to constant factors).

The Euclidean Steiner Tree Problem: Simulated Annealing and Other Heuristics

<p>In this thesis the Euclidean Steiner tree problem and the optimisation technique called simulated annealing are studied. In particular, there is an investigation of whether simulated annealing is a viable solution method for the problem. The Euclidean Steiner tree problem is a topological network design problem and is relevant to the design of communication, transportation and distribution networks. The problem is to find the shortest connection of a set of points in the Euclidean plane. Simulated annealing is a generally applicable method of finding solutions of combinatorial optimisation problems. The results of the investigation are very satisfactory. The quality of simulated annealing solutions compare favourably with those of the best known tailored heuristic method for the Euclidean Steiner tree problem</p>

The Euclidean Steiner Tree Problem: Simulated Annealing and Other Heuristics

<p>In this thesis the Euclidean Steiner tree problem and the optimisation technique called simulated annealing are studied. In particular, there is an investigation of whether simulated annealing is a viable solution method for the problem. The Euclidean Steiner tree problem is a topological network design problem and is relevant to the design of communication, transportation and distribution networks. The problem is to find the shortest connection of a set of points in the Euclidean plane. Simulated annealing is a generally applicable method of finding solutions of combinatorial optimisation problems. The results of the investigation are very satisfactory. The quality of simulated annealing solutions compare favourably with those of the best known tailored heuristic method for the Euclidean Steiner tree problem</p>