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Accelerating Computation of Steiner Trees on GPUs

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Abstract

The Steiner Tree Problem (STP) is a well studied graph theoretic problem. It computes a minimum-weighted tree of a given graph such that the tree spans a given subset of vertices called terminals. STP is NP-hard. Due to its wide applicability, it has been a challenge problem in the 11th DIMACS implementation challenge and the PACE 2018 challenge. Due to its importance, polynomial-time approximation algorithms have been devised for solving the STP. One of the most popular algorithms is by Kou, Markowsky and Berman (KMB) which provides a 2-approximation to STP. In practice, a naïve implementation of the KMB algorithm is prohibitively slow for large graphs. Our goal in this work is to improve KMB algorithm’s practical utility by parallelizing it on GPU and reduce its execution time on real-world graphs. This parallelization faces several challenges due to the inherent irregular nature of computation, as well as critical design decisions affecting the algorithm choice and optimizations. We overcome these challenges with interesting algorithmic observations and by exploiting parallelization within sub-steps, and develop the first GPU-based efficient approach to computing Steiner trees using KMB. We illustrate the effectiveness of our approach using several graph benchmarks from the DIMACS Challenge, the PACE Challenge, SteinLib, and SNAP. Our highly optimized GPU implementation achieves an average 20\(\times\) speedup over the CPU-sequential Open Graph algorithms and Data structure (OGDF)’s KMB implementation. In addition to this, our optimized CPU implementation achieves an average 4\(\times\) over OGDF’s KMB, the only published open-source KMB implementation.

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Notes

  1. One may further explore algorithmic variants such as \(\varDelta\)-stepping, which may suit their setup.

  2. Diameter is the maximum eccentricity. Eccentricity(v) is the largest distance from v to all the vertices in V, i.e., d(v, V). Here, diameter over terminal means \(\max d(v,L)\).

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Acknowledgements

We thank the anonymous reviewers for suggesting to implement p-SSSP. This has resulted in improved performance. We thank PACE 2018 PC members Florian Sikora and Édouard Bonnet for sharing the private instances of the challenge after the results and optil.io point of contact Jan Badura for enabling us to submit on optil.io post PACE related to this publication. Thanks to Stephan Beyer for rightly pointing us to use OGDF for the source codes of their JEA paper.

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  1. Department of Computer Science and Engineering, IIT Madras, Chennai, Tamil Nadu, India

    Rajesh Pandian Muniasamy, Rupesh Nasre & N. S. Narayanaswamy

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Correspondence to Rajesh Pandian Muniasamy.

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The second author is supported by the NSM project CS/19-20/1123/MEIT/008606.

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Muniasamy, R.P., Nasre, R. & Narayanaswamy, N.S. Accelerating Computation of Steiner Trees on GPUs. Int J Parallel Prog 50, 152–185 (2022). https://doi.org/10.1007/s10766-021-00723-0

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