# Dynamic Programming for Minimum Steiner Trees

@article{Fuchs2007DynamicPF, title={Dynamic Programming for Minimum Steiner Trees}, author={Bernhard Fuchs and Walter Kern and Daniel M{\"o}lle and Stefan Richter and Peter Rossmanith and Xinhui Wang}, journal={Theory of Computing Systems}, year={2007}, volume={41}, pages={493-500} }

We present a new dynamic programming algorithm that solves the minimum Steiner tree problem on graphs with k terminals in time O*(ck) for any c > 2. This improves the running time of the previously fastest parameterized algorithm by Dreyfus-Wagner of order O*(3k) and the so-called "full set dynamic programming" algorithm solving rectilinear instances in time O*(2.38k).

## 72 Citations

### Faster algorithm for optimum Steiner trees

- Computer ScienceInf. Process. Lett.
- 2011

### Parameterized Approximation Algorithms for Bidirected Steiner Network Problems

- Computer Science, Mathematics
- 2021

The Directed Steiner Network (DSN) problem takes as input a directed graph G=(V, E) with non-negative edge-weights and a set D V V of k demand pairs. The aim is to compute the cheapest network N G ...

### Parameterized Complexity of Arc-Weighted Directed Steiner Problems

- Computer Science, MathematicsSIAM J. Discret. Math.
- 2009

Three fundamental network design problems on arc-weighted directed graphs are investigated with respect to the parameters "number of terminals", "an upper bound on the size of the connecting network", and the combination of both.

### Dijkstra meets Steiner: a fast exact goal-oriented Steiner tree algorithm

- Computer ScienceMath. Program. Comput.
- 2017

The algorithm matches the best known worst-case run time and has a fast, often superior, practical performance: on some large instances originating from VLSI design, previous best run times are improved upon by orders of magnitudes.

### An Integral Linear Programming Relaxation for Parameterized Steiner Trees

- Mathematics, Computer Science
- 2019

A novel linear program (LP) for the Steiner Tree problem, where a set of terminal vertices need to be connected by a minimum weight tree in an edge-weighted graph, which shows that the problem admits an integral LP with 3|E| variables and 2|V | constraints.

### Exact algorithms for the Steiner Tree Problem

- Computer Science
- 2008

This thesis improves the running time of algorithm to O(2.684k) by showing that the optimum Steiner tree T can be partitioned into T = T1 [ T2 [ T3 in a certain way such that each Ti is a minimum Steiners tree in a suitable contracted graph Gi with less than k 2 terminals.

### Title : Parameterized Single-Exponential Time Polynomial Space Algorithm for Steiner Tree Year : 2015 Version : Post print

- Computer Science, Mathematics
- 2016

This work gives the first single-exponential time, polynomial-space FPT algorithm for the weighted Steiner Tree problem, and gives an algorithm with running time O(7.97 ·n · logW ) using O(n · lognW · log k) space.

### A Fast Heuristic Algorithm for the Steiner Tree Problem with Uniform Edge Weights

- Computer Science2022 5th International Conference on Pattern Recognition and Artificial Intelligence (PRAI)
- 2022

An effective iterative local search algorithm for solving the Steiner tree problem in a graph with uniform edge weights using several strong graph reduction techniques and a novel neighborhood generation method, namely Steiner-edge-vertex swap is studied.

### Computing Optimal Steiner Trees in Polynomial Space

- Computer Science, MathematicsAlgorithmica
- 2012

This paper presents a O(1.55n)-time polynomial-space algorithm for the cardinality version of the Steiner tree problem, where all edge weights are one, and a improved branching strategy based on a improved branch strategy.

### On Extended Formulations For Parameterized Steiner Trees

- Mathematics, Computer ScienceIPEC
- 2021

It is proved that Steiner Tree admits an integral LP with O (3 k | E | ) variables and constraints, which matches the runtime of the Dreyfus-Wagner algorithm, and the poof gives a polyhedral perspective on this classic algorithm.

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An algorithm for solving the Steiner problem on a finite undirected graph is presented. This algorithm computes the set of graph arcs of minimum total length needed to connect a specified set of k…

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These algorithms — which are based on the generation and concatenation of full Steiner trees — are much more efficient than other approaches and allow exact solutions of problem instances with more than 2000 terminals.

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For random problem instances, this paper improves the theoretical worst-case time bound to O(n 2.38) and proves a running time of less than 2 " ; implementations show drastic improvements.

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An algorithm that computes an optimal rectilinear Steiner minimal tree of n points in at most O(n 2 2:62 n) time is presented, which is provably faster than any previous algorithm, and improves the previous best bound for practically solvable instances.

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The theoretical worst-case time bound to O(n2 . . . 2.38n) is improved, for random problem instances the authors prove a running time of αn with a constant α < 2 , which is an enormous improvement for exponential-time algorithms.

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The problem of determining the minimum length of an optimum rectilinear Steiner tree for a set A of points in the plane is shown to be NP-complete and the emphasis of the literature on heuristics and special case algorithms is well justified.

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We consider Steiner minimal trees in the plane with rectilinear distance. The rectilinear distance $d(p_1 ,p_2 )$ between two points $p_1 $, $p_2 $ is $| {x_1 - x_2 } | + | {y_1 - y_2 } |$, where the…

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The search for good approximation algorithms for the Steiner tree problem in graphs is to find algorithms that have polynomial running time and return solutions that are not far from an optimum solution.