# On the Geometry of Stable Steiner Tree Instances

@inproceedings{Freitag2021OnTG, title={On the Geometry of Stable Steiner Tree Instances}, author={James Freitag and Neshat Mohammadi and Aditya Potukuchi and L. Reyzin}, booktitle={Canadian Conference on Computational Geometry}, year={2021} }

In this work we consider the Steiner tree problem under Bilu-Linial stability. We give strong geometric structural properties that need to be satisﬁed by stable instances. We then make use of, and strengthen, these geometric properties to show that 1 . 59-stable instances of Euclidean Steiner trees are polynomial-time solvable by showing it reduces to the minimum spanning tree problem. We also provide a connection between certain approximation algorithms and Bilu-Linial stability for Steiner…

## References

SHOWING 1-10 OF 18 REFERENCES

### The Steiner tree problem on graphs: Inapproximability results

- MathematicsTheor. Comput. Sci.
- 2008

### On the Complexity of the Metric TSP under Stability Considerations

- Computer ScienceSOFSEM
- 2011

It is shown that for γ ≥ 1.8 a simple greedy algorithm computes the optimum Hamiltonian tour for every γ-stable instance of the Δ-TSP, whereas a simple local search algorithm can fail to find the optimum even if γ is arbitrary.

### Are Stable Instances Easy?

- Computer ScienceCombinatorics, Probability and Computing
- 2012

The notion of a stable instance for a discrete optimization problem is introduced, and it is argued that in many practical situations only sufficiently stable instances are of interest, and that this is indeed the case for the Max-Cut problem.

### Bilu-Linial Stable Instances of Max Cut and Minimum Multiway Cut

- Computer Science, MathematicsSODA
- 2014

It is proved that there is no robust polynomial-time algorithm for γ-stable instances of Max Cut when γ < α SC(n/2), where αSC is the best approximation factor for Sparsest Cut with non-uniform demands, and it is shown that the standard SDP relaxation for Max Cut is integral if [EQUATION].

### Clustering under Perturbation Resilience

- Computer ScienceSIAM J. Comput.
- 2016

This paper presents an algorithm that can optimally cluster instances resilient to $(1 + \sqrt{2})$-factor perturbations, solving an open problem of Awasthi et al.

### Reducibility Among Combinatorial Problems

- Computer Science50 Years of Integer Programming
- 1972

Throughout the 1960s I worked on combinatorial optimization problems including logic circuit design with Paul Roth and assembly line balancing and the traveling salesman problem with Mike Held, which made me aware of the importance of distinction between polynomial-time and superpolynomial-time solvability.

### A Framework for Algorithm Stability and Its Application to Kinetic Euclidean MSTs

- Computer ScienceLATIN
- 2018

Stability in general plays an important role in a wide variety of areas, such as numerical analysis, machine learning, and topology, but is poorly understood in the context of (combinatorial) algorithms.