# Better approximation bounds for the network and Euclidean Steiner tree problems

@inproceedings{Zelikovsky1996BetterAB, title={Better approximation bounds for the network and Euclidean Steiner tree problems}, author={Alex Zelikovsky}, year={1996} }

The network and Euclidean Steiner tree problems require a shortest tree spanning a given vertex subset within a network G=(V,E,d) and Euclidean plane, respectively. For these problems, we present a series of heuristics finding approximate Steiner trees with performance guarantees coming arbitrary close to 1+ln 2= 1.693... and 1+ln(2/sqrt3) = 1.1438..., respectively. The best previously known corresponding values are close to 1.746 and 1.1546.

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## 110 Citations

New Approximation Algorithms for the Steiner Tree Problems

- Computer Science, MathematicsJ. Comb. Optim.
- 1995

New approximation algorithms for the Steiner tree problems are designed using a novel technique of choosing Steiner points in dependence on the possible deviation from the optimal solutions.

A New Approximation Algorithm for the Steiner Tree Problem with Performance Ratio 5/3

- Mathematics, Computer ScienceJ. Algorithms
- 2000

This approach gives rise to conceptually much easier and faster (though randomized) sequential approximation algorithms for the Steiner tree problem than the currently best known algorithms from Karpinski and Zelikovsky which almost match their approximation factor.

An Efficient Approximation Algorithm for the Steiner Tree Problem

- Computer Science, MathematicsProceedings of the 2019 2nd International Conference on Information Science and Systems
- 2019

This article presents an efficient two-phase heuristic in greedy strategy that achieves an approximation ratio of 1.4295 and shows it is possible to achieve the same approximation guarantee while only solving hypergraphic LP relaxation once.

Lower bounds for the relative greedy algorithm for approximating Steiner trees

- Computer ScienceNetworks
- 2006

The lower bound for the performance ratio of the relative greedy algorithm is improved to 1.385, which is close to the upper bound of 1.694 provided in 1996.

A series of approximation algorithms for the acyclic directed steiner tree problem

- Mathematics, Computer ScienceAlgorithmica
- 2006

This paper gives anO(kε)-approximation algorithm for any ε>0.1, which improves the previously knownk-approximating.

A 1.376 Approximation Algorithm for the Steiner Tree Problem

- Mathematics
- 2017

The Steiner tree problem is one of the classic and most fundamental NP-hard problems: given an arbitrary weighted graph, seek a minimum-cost tree spanning a given subset of the vertices (terminals).…

Lower Bounds for Approximation Algorithms for the Steiner Tree Problem

- Mathematics, Computer ScienceWG
- 2001

This paper provides for several Steiner tree approximation algorithms lower bounds on their performance ratio that are much larger, and proves lower bounds that match the upper bounds of these algorithms.

Tighter Bounds for Graph Steiner Tree Approximation

- Mathematics, Computer ScienceSIAM J. Discret. Math.
- 2005

A new polynomial-time heuristic is presented that achieves a best-known approximation ratio of 1 + \frac{\ln 3}{2} \approx 1.55$ for general graphs and best- known approximation ratios of 1.28 for both quasi-bipartite graphs and complete graphs with edge weights 1 and 2.

Lower bounds for the relative greedy algorithm for approximating Steiner trees

- Mathematics
- 2006

The Steiner tree problem is to find a shortest subgraph that spans a given set of vertices in a graph. This problem is known to be NP-hard, and it is well known that a polynomial time 2-approximation…

An Improved Approximation Algorithm for the Terminal Steiner Tree Problem

- Mathematics, Computer ScienceICCSA
- 2011

This paper presents an approximation algorithm with performance ratio 2ρ - (ρα2 - αρ)/(α+α2)(ρ-1)+2(α-1)2 for the terminal Steiner tree problem, where ρ is the best-known performance ratio for the Steiner Tree problem with any α ≥ 2.

## References

SHOWING 1-10 OF 14 REFERENCES

The Steiner Problem with Edge Lengths 1 and 2

- Mathematics, Computer ScienceInf. Process. Lett.
- 1989

A 4 3 -approximation algorithm is given for the special case in which the underlying network is complete and all edge lengths are either 1 or 2, showing that this special case is MAX SNP-hard, which may be evidence that the Steiner problem on networks has no polynomial-time approximation scheme.

Improved approximations for the Steiner tree problem

- Mathematics, Computer ScienceSODA '92
- 1992

An approximation algorithm is given and it is shown that the worst-case ratio of the cost of the solutions to the optimal cost is better than previously known ratios in graphs, and in rectilinear metric on the plane.

Further improvements of Steiner tree approximations

- Mathematics
- 1994

The Steiner tree problem requires to find a shortest tree eonneeting a given set of terminal points in ametrie space. We suggest a better and fast heuristie for the Steiner problem in graph! and in…

On better heuristic for Euclidean Steiner minimum trees

- Mathematics, Computer Science[1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science
- 1991

It is shown that there exists a polynomial-time heuristic with a performance ratio bigger than square root 3/2.

The computation of nearly minimal Steiner trees in graphs

- Mathematics
- 1983

The computation of a minimal Steiner tree for a general weighted graph is known to be NP‐hard. Except for very simple cases, it is thus computationally impracticable to use an algorithm which…

Worst-Case Performance of Rayward-Smith's Steiner Tree Heuristic

- Mathematics, Computer ScienceInf. Process. Lett.
- 1988

Abstract In this paper we prove that the worst-case performance of the Steiner tree approximation algorithm by Rayward-Smith (RS) is within two times optimal and that two is the best bound in the…

An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms

- Mathematics, Computer Science[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science
- 1988

The main result is an algorithm for performing the task provided that the capacity of each cut exceeds the demand across the cut by a Theta (log n) factor.

An approach for proving lower bounds: solution of Gilbert-Pollak's conjecture on Steiner ratio

- Mathematics, Computer ScienceProceedings [1990] 31st Annual Symposium on Foundations of Computer Science
- 1990

A long-standing problem about Steiner minimum trees and minimum spanning trees is solved and it is proved that, for any P, L/sub S/(P)>or= square root 3L/sub m/(P)/2, as conjectured by E.N. Gilbert and H.O. Pollak.

A Limited-Backtrack Greedy Schema for Approximation Algorithms

- Computer ScienceFSTTCS
- 1994

This paper studies algorithmic schemas of the following kind: at each stage, the algorithm can either take one new step, or can undo a previous step and take two new steps, whichever results in the largest gain in the objective function value.

Proof verification and hardness of approximation problems

- Computer Science, MathematicsProceedings., 33rd Annual Symposium on Foundations of Computer Science
- 1992

The authors improve on their result by showing that NP=PCP(logn, 1), which has the following consequences: (1) MAXSNP-hard problems do not have polynomial time approximation schemes unless P=NP; and (2) for some epsilon >0 the size of the maximal clique in a graph cannot be approximated within a factor of n/sup ePSilon / unless P =NP.